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\title [\it Set theoretic real analysis ]
{\vspace {1.0cm}\uppercase 
{\Large \bf {Set Theoretic Real Analysis}}}
\author[K. Ciesielski]{\uppercase {\bf K. Ciesielski$^*$}}
%\address{}
\keywords{Fubini theorem, continuous images, continuous restrictions,
Continuum Hypothesis, Martin's Axiom, Blumberg Theorem, symmetric continuity, 
Peano curve, Darboux functions, cardinal functions, addition of functions,
derivatives, extensions of measure}
\subjclass{Primary: 26--02, 04--02; secondary 54--02, 28--02}
\thanks{$^*$The author wishes to thank Professors Lee Larson, 
Marek Balcerzak, Andy Bruckner, Tomasz Natkaniec, Juris Stepr\={a}ns, and
Brian Thomson 
for reading preliminary versions of this paper
and helping in improving its final version.
\newline
This work was partially supported by 
 NSF Cooperative Research Grant INT-9600548
with its Polish part financed by KBN.}
\thanks{\ \\
ISSN 1425--6908 \quad \copyright Heldermann Verlag}
\dedicatory{\footnotesize {\it Received February 25, 1997 and, 
in revised form, September 8, 1997}}
%\date{February 25, 1997}

\begin{document}
\leftline{\footnotesize \sc Journal of Applied Analysis}
\leftline{\footnotesize Vol. 3,  No. 2 (1997), pp. 143--190}
\vspace{3cm}
\centerline{\it Special invited paper}
\maketitle
\thispagestyle{empty}

\bigskip
\begin{abstract}
This article is a survey of the recent results that concern
real functions (from $\real^n$ into $\real$) and whose solutions or statements
involve the use of set theory.  The choice of the topics follows the author's
personal  interest in the subject, and there are probably some important results
in this area that did not make it to this survey.  Most of the results presented
here are left without proofs. 
\end{abstract}

\bigskip
\section*{\bf 1. Historical background}

\medskip
The study of real functions has played a fundamental role in the development of
mathematics over the last three centuries. 
The seventeenth century discovery of calculus by 
Newton and Leibniz was largely due
to increased understanding of the behavior of real functions. 
%The discovery of calculus by
%eighteenth century mathematicians, notably Newton and Leibniz, was largely due
%to increased understanding of the behavior of real functions. 
The birth of
analysis is often traced to the early nineteenth century work of Cauchy, who
gave precise definitions of concepts such as continuity and limits for real
functions. Convergence problems while approximating real functions by Fourier
series gave rise to both the Riemann and Lebesgue integrals. Cantor  developed
his set theory in an effort to answer uniqueness questions about Fourier series
\cite{KL,BKR,Th}.

During this time, different techniques have been used as the theory behind them
became available. For example, after Cauchy, various limiting operations such as
pointwise and uniform convergence were studied, giving rise to various
approximation techniques. At the turn of this century, measure theoretic
techniques were exploited, leading to stochastic convergence ideas in the
1920's.  Also, at about the same time topology was developed, and its
applications to analysis gave rise to functional analysis. 

In recent years, a new research trend has appeared which indicates the emergence
of a yet another branch  of inquiry that could be called {\em set theoretic real
analysis}.
This area is the study of families of real functions using
modern techniques of set theory. These techniques include advanced forcing
methods, special axioms of set theory such as Martin's axiom~(MA) and 
proper forcing axiom~(PFA), as well as some of their weaker consequences  like 
additivity of measure and category.   (See \cite{Ku}, \cite{Sh}, \cite{Fr},
\cite{DalWood}, and \cite{BJ} for examples of this work.) 

Set theoretic real analysis is closely allied with descriptive set theory, but
the objects studied in the two areas are different. The objects studied in
descriptive set theory are various classes of (mostly nice) {\bf sets} and their
hierarchies, such as Borel sets or analytic sets.  Set theoretic real analysis
uses the tools of modern set theory to study {\bf real functions} and is
interested mainly in more pathological objects.  Thus, the results concerning
subsets of the real line (like the series of studies on ``small'' subsets of
$\real$ 
\cite{AMillerSubsetsOfR}, or  deep studies of the duality between measure and
category 
\cite{Ox,AMiller:MeasureCat,BJ}) are considered only remotely related to the
subject.  (However, some of these duality studies spread to real analysis too.
For example, see a monograph~\cite{CLO:book}.)

Set theoretic real analysis already has a long history. Its roots can be traced
back to the 1920's, where powerful new techniques based on the Axiom of
Choice~(AC)  and the Continuum Hypothesis~(CH) can be seen in many papers from
such journals as Fundamenta Mathematicae and Studia Mathematica. 
The most interesting consequences of the Continuum Hypothesis
discovered in this period have been collected in~1934 monograph
of Sierpi\'nski, {\it Hypoth\`ese du Continu}~\cite{Si:CH}.
The influence of Sierpi\'nski's results (and the monograph) on the set 
theoretic real analysis can be best seen in the next section. 

The new
emergence of the field was sparked by the discovery of powerful new techniques
in set theory and can be compared to  the parallel development of set theoretic
topology during the late 1950's and 1960's. In fact, it is a bit surprising that
the development of set theoretic analysis is so much behind that of set
theoretic topology, since at the beginning of the century the applicability of
set theory in  analysis was at least as intense as in topology. This, however,
can be  probably attributed to the simple fact, that in the past half of a
century there were many mathematicians that knew well both topology and set
theory, and very few that knew  well simultaneously analysis and set theory. 

Our terminology is standard and follows ~\cite{Ci}.
\medskip
\stepcounter{section}
\section*{\bf 2. New developments in classical problems}

\medskip
The first problem we wish to mention
here is connected with the Fubini--Tonelli
Theorem. The theorem says, in particular, that if a function \linebreak
$f\colon[0,1]^2 \to[0,1]$ is measurable then  the iterated integrals 
$\int_0^1\int_0^1 f(x,y)\,dy\,dx$ and 
 $\int_0^1\int_0^1 f(x,y)\,dx\,dy$  exist
and are both equal to the double integral \linebreak 
$\int\int f\, dm_2$, where $m_2$
stands for the Lebesgue measure on $\real^2$.  But what happens when $f$ is
non-measurable?  Then clearly the double integral does not exist. However, the
iterated integrals might still  exist. Must they be equal? The next theorem,
which is a classical example of an application of the Continuum Hypothesis in
real analysis, gives a negative answer to this question.

\medskip
\begin{theorem}[Sierpi\'nski 1920, \cite{Si1}] %2.1
\label{th:sierp1}
 If the Continuum
Hypothesis holds then there  exists %a function 
$f\colon[0,1]^2\to[0,1]$  for which the iterated integrals 
$\int_0^1\int_0^1 f(x,y)\,dy\,dx$ and $\int_0^1\int_0^1 f(x,y)\,dx\,dy$  exist
but are not equal.
\end{theorem}

\medskip
\begin{pf}  Let $\preceq$ be a well ordering of $[0,1]$ in order type  continuum
$\continuum$ and define $A=\{\la x,y\ra\in[0,1]^2\colon x\preceq y\}$. Let $f$
be the characteristic function $\charf{A}$ of $A$.  Then for every fixed
$y\in[0,1]$ the set
$\{x\in[0,1]\colon f(x,y)\neq 0\}= \linebreak
\{x\in[0,1]\colon x\preceq y\}$  is an initial
segment of a set ordered in type $\continuum$. So, by CH, it is at most
countable and
\[
\int_0^1\int_0^1 f(x,y)\,dx\,dy=\int_0^1 0\,dy=0.
\] 
Similarly, for each $x\in[0,1]$ the set
$\{y\in[0,1]\colon f(x,y)\neq 1\}=\linebreak 
\{y\in[0,1]\colon y\prec x\}$ is at most
countable and
\[
\int_0^1\int_0^1 f(x,y)\,dy\,dx=\int_0^1 1\,dy=1.
\] 
Thus, 
$\int_0^1\int_0^1 f(x,y)\,dy\,dx=1\neq 0=\int_0^1\int_0^1 f(x,y)\,dx\,dy$.
\end{pf}

\medskip
Sierpi\'nski's use of the Continuum Hypothesis in the construction of such a
function begs the question whether such a function can be constructed using only
the axioms of ZFC. The negative answer was given in the 1980's by 
Laczkovich~\cite{La}, Friedman~\cite{Fd} and Freiling~\cite{Fl},  who
independently proved the following theorem.

\medskip
\begin{theorem}%2.2
\label{th:sierp1NEG}
There exists a model of set theory ZFC  in which for every
function $f\colon[0,1]^2\to[0,1]$, the existence of the iterated integrals 
$\int_0^1\int_0^1 f(x,y)\,dy\,dx$ and $\int_0^1\int_0^1 f(x,y)\,dx\,dy$  implies
their equality.
\end{theorem}

\medskip
It is also worthwhile to mention that the function $f$ from the proof of
Theorem~\ref{th:sierp1NEG} has the desired property as long as every subset of
$\real$ of cardinality less than continuum has measure zero, i.e., when the
smallest cardinality $\non(\calN)$ of the non-measurable subset of $\real$ is
equal to $\continuum$. Since the equation $\non(\calN)=\continuum$ holds in many
models of ZFC in which CH fails (for example, it is implied by MA)
Theorem~\ref{th:sierp1NEG} is certainly not equivalent to CH. On the other hand,
Laczkovich proved Theorem~\ref{th:sierp1NEG} by noticing that:  (A) the
existence of an example as in the statement of Theorem~\ref{th:sierp1}
implies the existence of such an example as  in its proof, i.e., in form of
$\charf{A}$;  (B) there is no set $A\subset[0,1]^2$ with $f=\charf{A}$
satisfying Theorem~\ref{th:sierp1NEG} if 
$\non(\calN)<\cov(\calN)$, where $\cov(\calN)$ is  the smallest cardinality of a
covering of $\real$ by the sets of measure zero.  (It is well known that the
inequality 
$\non(\calN)<\cov(\calN)$ is consistent with ZFC.
See e.g.~\cite{BJ}.)

A discussion of a similar problem for the functions 
$f\colon[0,1]^n\to[0,1]$ and the $n$-times iterated integrals can be found 
in a~1990
paper of Shipman~\cite{Ship}. The same paper contains also 
two easy ZFC examples of measurable functions
$f\colon [0,1)^2\to\real$ and $g\colon \real^2\to[-1,1]$
for which the iterated integrals exist but are not equal.
Thus, the restriction of the above problem to 
the non-negative functions is essential. 

\medskip
Another classical result arises from a different theorem of 
Sierpi\'nski of~1928.

\medskip
\begin{theorem}[Sierpi\'nski \cite{Si2a,Si2b}] %2.3
\label{th:sierp2}
 If the Continuum Hypothesis \linebreak 
holds then there exists a set
$S\subset\R$ of cardinality continuum such that  its image $f[S]\neq [0,1]$ for
any continuous $f\colon\R\to[0,1]$.
\end{theorem}

\medskip
The set $S$ from the original proof of Theorem~\ref{th:sierp2} is 
called {\em Sierpi\'nski set}\/ and it has the property that  its intersection
$S\cap N$ with any measure zero set $N$  is at most countable.%
\footnote{This approach was used in the paper~\cite{Si2a}, 
while the Luzin set
approach in the paper~\cite{Si2b}. 
Since they are published in the same year,
the priority is not completely clear. However in the list of Sierpi\'nski's
publications  printed in~\cite{Si:Colect} paper~\cite{Si2a} precedes~\cite{Si2b},
suggesting its priority.} 
Another set that satisfies the conclusion of
Theorem~\ref{th:sierp2}, known as {\em Luzin set}\/ (see~\cite{Si2b}
or~\cite[property C$_5$]{Si:CH}),  is defined as an uncountable subset $L$ of
$\real$ whose intersection
$L\cap M$ with any meager set $M$ is at most countable.%
\footnote{The construction of such a set, under CH,
was published by Luzin in~1914~\cite{LusSet}.
The same construction had been also published in 1913 by Mahlo~\cite{Mahlo}.
But (as is not unusual in mathematics) such a set is commonly known as 
a Luzin set.}
The existence of a
Luzin set is also implied by CH.  In fact, the constructions of sets $S$ and $L$ 
under the assumption of CH are almost identical: you list all G$_\delta$ measure
zero sets  (F$_\sigma$ meager sets) as $\{Z_\xi\colon \xi<\continuum\}$ and
define $S$ ($L$, respectively) as a set $\{x_\alpha\colon\alpha<\continuum\}$
where 
$x_\alpha\in\real\setminus\left(\bigcup_{\xi<\alpha}Z_\xi\right)$. The choice is
possible since, by CH, the family
$\{Z_\xi\colon\xi<\alpha\}$ is at most countable implying that its union is not
equal to $\real$.

It is also easy to see that this construction can be carried out if
$\cov(\N)=\continuum$ (and its category analog $\cov(\M)=\continuum$ in case of
construction of $L$). The sets constructed that way are called generalized
Sierpi\'nski and Luzin sets, respectively, and they also satisfy the conclusion
of Theorem~\ref{th:sierp2} independently of the size of $\continuum$. Since 
many models of ZFC satisfy either 
$\cov(\N)=\continuum$ or $\cov(\M)=\continuum$ (for example, both conditions are
implied by MA) it has been a difficult task to find a model of ZFC in which the
conclusion of Theorem~\ref{th:sierp2} fails. It has been found by A.~W.~Miller
in~1983.

\medskip
\begin{theorem}[A.~W.~Miller \cite{Mi}]%2.4
\label{th:sierp2NEG}
 There exists a model of set
theory ZFC in which for every  subset $S$ of\/ $\real$ of cardinality
$\continuum$ there  exists a continuous function $f\colon\R\to[0,1]$ such that
$f[S]=[0,1]$.
\end{theorem}

\medskip
In his proof of Theorem~\ref{th:sierp2NEG} Miller used  the iterated perfect set
model, which will be mentioned in this paper in several other occasions. 

\medskip
Some of the most recent set-theoretic results  concerning classical problems in
real functions  are connected with a theorem of Blumberg from~1922.

\medskip
\begin{theorem}[Blumberg \cite{Bl}] %2.5
\label{th:Blumb}  
For every $f\colon\R\to\R$ there
exists a dense subset
$D$ of\/ $\R$ such that the restriction 
$f\restriction D$ of $f$ to $D$ is continuous.
\end{theorem}

\medskip
The set $D$ constructed by Blumberg is countable.  In a quest whether it can be
chosen any bigger  Sierpi\'nski and Zygmund proved in~1923 the following theorem.

\medskip
\begin{theorem}[Sierpi\'nski, Zygmund \cite{SZ}]%2.6
\label{th:SZ}  
There exists a function 
$f\colon\R\to\R$ whose restriction $f\restriction X$ is discontinuous for any 
subset $X$ of\/ $\R$ of cardinality $\continuum$.
\end{theorem}

\medskip
Theorem~\ref{th:SZ} immediately implies the following corollary, which shows
that there is no hope for proving in ZFC a version of the Blumberg theorem in
which the set $D$ is uncountable.

\medskip
\begin{corollary}[Sierpi\'nski, Zygmund \cite{SZ}] %2.7
\label{cor:SZ}
If the Continuum Hy\-po\-the\-sis holds 
then there  exists a function $f\colon\R\to\R$ 
such that $f\restriction X$ is 
discontinuous for any uncountable  subset $X$ of\/ $\R$.
\end{corollary}

\medskip
The proof of Theorem~\ref{th:SZ} is a straightforward transfinite induction
diagonal argument after noticing that every continuous  partial function on $\R$
can be extended to a continuous  function on a G$_\delta$ set. 

Corollary~\ref{cor:SZ} raises the natural question about the importance of the
assumption of CH in its statement. Is it consistent that the set $D$ in Blumberg
Theorem can be uncountable? Can it be of positive outer measure, or non-meager? 

The cardinality part of these questions is addressed by the following theorem of
Baldwin from~1990.

\medskip
\begin{theorem}[Baldwin \cite{Ba}] %2.8
\label{th:Baldwin}
If Martin's Axiom holds then for
every function $f\colon\R\to\R$  and every infinite cardinal number
$\kappa<\continuum$  there exists a set $D\subset\R$ such that 
$f\restriction D$ is continuous and $D$ is $\kappa$-dense, i.e.,
$D\cap I$ has cardinality $\kappa$ for every non-trivial interval $I$.
\end{theorem}

\medskip
Thus under MA the size of the set $D$ is clear. By Theorem~\ref{th:SZ} it cannot
be chosen of cardinality continuum (at least for some functions), but it can be
always  chosen of any cardinality $\kappa$ less than $\continuum$. 

One might still hope to be able to prove in ZFC that  for any $f$ the set $D$
can be found of an arbitrary cardinality $<\continuum$. However, this is false
as well, as noticed by several authors: 
G.~Gruenhage in~1993 (see Rec{\l}aw~\cite[Thm~4]{Recl})
S.~Shelah in~1995 (see~\cite{Sh2}) and the author of this survey
(unpublished). 

\medskip
\begin{theorem}[\cite{Recl,Sh2}] %2.9
\label{th:Sh1} 
There exists a model of ZFC+$\neg$CH
(namely a Cohen model) in which there is a function $f\colon\real\to\real$ which
is discontinuous on any uncountable subset of\/ $\R$.
\end{theorem}

\medskip
The category version of a question on a size of $D$ has been  also settled
in the~1995 paper of Shelah~\cite{Sh2} mentioned above. 

\medskip
\begin{theorem}[Shelah \cite{Sh2}] %2.10
\label{th:Sh2}
There exists a model of ZFC in which for
every function $f\colon\real\to\real$ there exists a set $D\subset\R$ such that 
$f\restriction D$ is continuous and $D$ is nowhere meager, i.e.,
$D\cap I$ is non-meager for every non-trivial interval $I$.
\end{theorem}

\medskip
The measure version of the question is less clear. 
It has been noticed by J.~Brown in~1977 that 
the precise measure analog of
Theorem~\ref{th:Sh2} cannot be proved.
(This has been also noticed independently by K.~Ciesielski,
whose proof is included below.)

\medskip
\begin{theorem}[Brown \cite{BrownMeasBl}] %2.11
\label{th:KCBlumb}
 There exists a function
$f\colon\R\to\R$  such that $f\restriction D$ is discontinuous for every set
$D\subset\real$ which is nowhere measure zero, i.e., such that
$D\cap I$ has positive outer measure for every non-trivial interval $I$.
\end{theorem}

\medskip
\begin{pf} 
 Let $\{F_n\colon n<\omega\}$ be a partition of $\real$ such that
$F_0$ is a dense G$_\delta$ set of measure zero and 
$F_n$ is nowhere dense for each $n>0$.  Define $f\colon\real\to\real$ by putting
$f(x)=n$ for $x\in F_n$. Now, $f\restriction X$ is discontinuous for any dense
$X\subset\real$ which is nowhere measure zero. 

Indeed, if $X\subset\real$ is dense and nowhere measure zero then there exists
an $x\in X\setminus F_0$.  Now, if every open set $U$ containing $x$ intersects
$F_n\cap X$ for infinitely many $n$ then $f\restriction X$ is discontinuous at
$x$. Otherwise, there is an open set $U$ containing $x$ and intersecting only
finitely many $F_n$'s. So, we can find a non-empty open interval $I\subset U$
such that $I\cap X\subset F_0$. But this means that $I\cap X$ has measure zero,
a contradiction. 
\end{pf}

\medskip
However, the following problem asked by Heinrich von Weizs\"acker
\cite[Problem AR(a)]{FrProb96} remains open.

\medskip
\begin{problem} %1
\label{pr:Blum}
Is it consistent that every function $f\colon\real\to\real$ is
continuous on some set $X\subset\real$ of positive outer measure?
\end{problem}

\medskip
Other generalizations of Blumberg's theorem can be also found in 
a~1994 survey article~\cite{BrownRestr}. 
(See also recent papers~\cite{BrownNew} and~\cite{JRblum}.)

Another problem that is related in character to the Blumberg's theorem
is the following. 

\begin{quote}
{\it Let $\{f_n\colon\real\to[-\infty,\infty]\}_{n=1}^\infty$ be a sequence
of arbitrary functions. What is the biggest size of a set $X\subset\real$
for which there exists a subsequence of
$\{f_n\}_{n=1}^\infty$ convergent pointwise on $X$?}
\end{quote}
Clearly such a subsequence can be found for any countable $X\subset\real$.
Using this fact Helly~\cite{Hel} proved in~1921 that 
{\it any bounded sequence of 
monotone real functions contains a pointwise convergent subsequence}.
On the other hand, answering a question of S.~Saks,
in~1932 Sierpi\'{n}ski~\cite{Si1932}
showed that the Continuum Hypothesis implies
the existence of a sequence 
$\{f_n\colon\real\to[-\infty,\infty]\}_{n=1}^\infty$
such that 
$\{f_n\restriction X\}_{n=1}^\infty$ has no pointwise convergent subsequence 
for any uncountable $X\subset\real$.
The necessity of additional set theoretical assumptions in 
the Sierpi\'{n}ski's construction was recently noticed by
Fuchino and Plewik~\cite{FP} who showed
that the size of $X$ having the property under consideration is 
characterized by the splitting number ${\bold s}$:
{\it For any $X\subset\real$ with $|X|<{\bold s}$ any sequence
$\{f_n\colon\real\to[-\infty,\infty]\}_{n=1}^\infty$ has a subsequence convergent
pointwise on $X$; however for 
any $X\subset\real$ with $|X|={\bold s}$ there exists a sequence
$\{f_n\colon X\to[0,1]\}_{n=1}^\infty$ 
with no pointwise convergent subsequence.}
(For the definition of the splitting number, see e.g.~\cite{V}.)

\medskip
In the past few years a lot of activity in real analysis was concentrated around
symmetric properties of real functions. (See Thomson~\cite{Th}.) Recall that a
function $f\colon\real\to\real$ is symmetrically continuous at $x\in\real$ if
\[
\lim_{h\to 0}(f(x+h)-f(x-h))=0,
\] 
and $f$ is approximately symmetrically differentiable at $x$ if  there exists
a set $S\subset\real$ such that 
$x$ is a (Lebesgue) density point of $\real\setminus S$ and that the following
limit exists
\[
\lim_{h\to 0,\, h\notin S}\frac{f(x+h)-f(x-h)}{2h}.
\] This limit, which does not depend on the choice of a set $S$, is called 
{\em
the approximate symmetric derivative of $f$ at $x$}\/ and is denoted by
$D^s_{ap}f(x)$.  We will say that $f$ has a {\em co-countable symmetric
derivative at $x$}\/ and denote it by $D^s_c f(x)$ if the set $S$ in the above
definition  can be chosen to be countable. 

One of the long standing conjectures  (with several incorrect proofs given
earlier, some even published)  was settled by Freiling and Rinne in~1988 by
proving the following theorem.

\medskip
\begin{theorem}[Freiling, Rinne \cite{FR}] %2.12
\label{th:FR} 
If $f\colon\R\to\R$ is
measurable and such that
$D^s_{ap}f(x)=0$ for all $x\in\R$ then $f$ is constant almost everywhere.
\end{theorem}

\medskip
The importance of the measurability assumption in Theorem~\ref{th:FR} was long
known from the following theorem of Sierpi\'nski of~1936.

\medskip
\begin{theorem}[Sierpi\'nski \cite{Si4}] %2.13
\label{th:SiSymm}
If the Continuum Hypothesis
holds then there exists a non-measurable function
$f\colon\R\to\R$ (which is a characteristic function $\charf{A}$ of some set
$A$) for which $D^s_c f(x)=0$ for all $x\in\R$.
\end{theorem}

\medskip
In fact, in~\cite{Si4} Theorem~\ref{th:SiSymm} is 
stated in a bit stronger form%
\footnote{Co-countable symmetric derivatives are replaced by co-$<\continuum$
symmetric derivatives and the theorem is proved in ZFC.} from which it follows
immediately that the theorem remains true under MA, if the co-countable
symmetric derivatives  $D^s_c f(x)$ are replaced by  the approximate symmetric
derivatives $D^s_{ap}(x)$. However, neither Theorem~\ref{th:SiSymm} nor its
version with $D^s_{ap}(x)$ can be proved in ZFC. This follows from the following
two theorems of Freiling from~1990.

\medskip
\begin{theorem}[Freiling \cite{Frei:CC}] %2.14
\label{th:Freiling1}
If the Continuum Hypothesis
fails then for every function
$f\colon\R\to\R$  with $D^s_c f(x)=0$ for all $x\in\R$ there exists a countable
set $S$ such that $f$ is constant on $\real\setminus S$.
\end{theorem}

\medskip
Thus the existence of a function as in Theorem~\ref{th:SiSymm} is in fact
equivalent to the Continuum Hypothesis. 

\medskip
\begin{theorem}[Freiling \cite{Frei:CC}] %2.15
\label{th:Freiling2}
It is consistent with ZFC
that for every function
$f\colon\R\to\R$  with $D^s_{ap} f(x)=0$ for all $x\in\R$ there exists a measure
zero set $S$ such that $f$ is constant on $\real\setminus S$.
\end{theorem}

\medskip
More precisely, Freiling proves that the conclusion of 
Theorem~\ref{th:Freiling2} follows  the property that is just a bit stronger than
the inequality $\non(\calN)<\cov(\calN)$. (Compare comment following
Theorem~\ref{th:sierp1NEG}.)

\medskip
Another direction in which the symmetric continuity research went was the study
of how far symmetric continuity can be destroyed.  First note that clearly every
continuous function is symmetrically continuous, but not vice versa, since the
characteristic function of a singleton is symmetrically continuous. However, it
is not difficult  to find functions which are nowhere symmetrically continuous.
For example, the characteristic function of any dense Hamel basis is such a
function.%
\footnote{In fact, a Hamel basis $\B$ can be chosen to be both first category and
measure zero. Thus $\charf{\B}$ can be measurable and have the Baire property.}
How much more can we destroy symmetric continuity?

In the non-symmetric case probably the weakest (bilateral) version of 
continuity that can be defined is the following.  A function
$f\colon\real\to\real$ is {\em weakly continuous}\label{pageWC}  at $x$ if there
are sequences $a_n\nearrow0$ and $b_n\searrow0$ such that
\[
\lim_{n\to\infty}f(x+a_n)=f(x)=\lim_{n\to\infty}f(x+b_n).
\] This notion is so weak that it is impossible to find a function
$f\colon\real\to\real$ which is nowhere weakly continuous. This follows from the
following easy, but a little surprising theorem.

\medskip
\begin{theorem}[{\cite[p. 82]{CollingwoodLohwater:ThClSets}}]%2.16
\label{th:WeakC}
 Every function
$f\colon\real\to\real$ is weakly continuous everywhere on the complement of a
countable set.
\end{theorem}

\medskip
A natural symmetric counterpart of weak continuity is defined as follows.  A
function $f\colon\real\to\real$ is {\em weakly symmetrically continuous} at $x$
if there is a sequence $h_n\to 0$ such that 
\[
\lim_{n\to\infty}(f(x+h_n)-f(x-h_n))=0.
\] 
However, the symmetric version of Theorem~\ref{th:WeakC} badly fails: there
exist nowhere weakly symmetrically continuous functions (which are also called
{\em uniformly antisymmetric functions}). Their existence follows immediately
from the following theorem of Ciesielski and Larson from~1993.

\medskip
\begin{theorem}[Ciesielski, Larson \cite{CL}]%2.17
\label{th:CLUnifA}
 There exists a function \linebreak
$f\colon\R\to\natural$ such that the set
\[ 
S_x=\{h>0\colon f(x+h)=f(x-h)\}
\] 
is finite for every $x\in\R$.
\end{theorem}

\medskip
The function $f$ from Theorem~\ref{th:CLUnifA} raises the questions in two
directions. Can the range of $f$ be any smaller? Can the size of all sets
$S_x$ be
uniformly bounded? The first of this questions leads to the following  open
problem from~\cite{CL}. (See also problems listed  in~\cite{Th}.)

\begin{problem} %2
\label{pr:CLUA}
 Does there exist a uniformly antisymmetric function \linebreak
$f\colon\real\to\R$ with range $f[\R]$ being (a) finite? (b) bounded?
\footnote{A uniformly antisymmetric function $f\colon \real \to [0,1]$
has been recently constructed by S. Shelah.}
\end{problem}

\medskip
Concerning part (a) of this problem it has been proved in~1993  by
Ciesielski~\cite{C0} that the range of a uniformly antisymmetric function must
have at least $4$ elements. (Compare also~\cite{C4}.)

The estimation of sizes of sets $S_x$ from Theorem~\ref{th:CLUnifA} has been
examined by Komj\'{a}th and Shelah in~1993, leading to the following two
theorems.

\medskip
\begin{theorem}[Komj\'{a}th, Shelah \cite{KomSh}]%2.18
\label{th:KomSh1}The Continuum
Hypothesis is equivalent to the existence of a function $f\colon\R\to\natural$
such that the set
\[ 
S_x=\{h>0\colon f(x+h)=f(x-h)\}
\] 
has at most $1$ element for every $x\in\R$.
\end{theorem}

\medskip
\begin{theorem}[Komj\'{a}th, Shelah \cite{KomSh}]%2.19
\label{th:KomSh2}
  If
$\continuum>\omega_{k+1}$, $k=0,1,2,\ldots$, then there is no function
$f\colon\R\to\natural$ such that the set
\[ 
S_x=\{h>0\colon f(x+h)=f(x-h)\}
\] 
has at most $2^k$ elements for every $x\in\R$.
\end{theorem}

\medskip
Theorem~\ref{th:KomSh1} suggests that the converse of  Theorem~\ref{th:KomSh2}
should also be true.  However, this is still unknown, leading to another open
problem.

\medskip
\begin{problem}%3
\label{pr:SandD}
Does the assumption that $\continuum\leq\omega_{k+1}$ imply that
there exists a function $f\colon\R\to\natural$ such that the set
\[ 
S_x=\{h>0\colon f(x+h)=f(x-h)\}
\] 
has at most $2^k$ elements for every $x\in\R$?
\end{problem}

\medskip
For $k=0$ the positive answer is implied by Theorem~\ref{th:KomSh1}. Also, it is
consistent that $\continuum=\omega_{k+1}$ and there exists
$f\colon\R\to\natural$ such that each
$S_x$ has at most $2^k$ elements.  This follows from another theorem of
Komj\'{a}th and Shelah
\cite[Thm 1]{KomSh2}. (See also a paper~\cite{C:SumsAndDiff} of Ciesielski
related to this subject.) 

In fact, the proof of Theorem~\ref{th:CLUnifA} gives also the following version
for functions on 
$\real^n$:
\begin{itemize}
\item There exists a function $f\colon\R^n\to\natural$ such that the set
\[
\{h\in\real^n\colon f(x+h)=f(x-h)\}
\] 
is finite for every $x\in\R^n$.
\end{itemize} 
This statement is related to the following  recent theorem of
J.~Schmerl, which solves  a long standing problem of
Erd\H{o}s~\cite[Problem~15.9]{millerq}. (See also a survey article~\cite{Kom}
for more on this problem.)

\medskip
\begin{theorem}[Schmerl~\cite{Schmerl}]%2.20
\label{th:Schmerl}
 There exists a function
$f\colon\R^n\to\natural$ such that for any distinct $a,b,x\in\real^n$ with
$\|a-x\|=\|x-b\|$ all the values $f(a)$, $f(x)$ and $f(b)$ are not equal. 
\end{theorem}

\medskip \noindent
Thus, this theorem says, that there exists (in ZFC) a countable partition  of
$\real^n$ such that no three vertices $a,b,x$ spanning isosceles  triangle
belong to the same element of the partition. 

\medskip
\stepcounter{section}
\section*{\bf 3. New classic-like results}

\medskip
Consider a function $F=\la f_1,f_2\ra$ from $\real$ onto $\real^2$. By a well
known theorem of Peano from~1890 (see e.g.~\cite{Sagan}) such an $F$ can be
continuous. However, it is not difficult to see that it cannot be
differentiable.  It follows easily from the fact that every differentiable
function
$f\colon\real\to\real$ satisfies the Banach condition $T_2$, i.e.,  the set
$\{y\colon f^{-1}(y)\text{ is uncountable}\}$ has Lebesgue measure zero. (See
e.g. \cite[Chap. VII, p. 221]{Saks}.) Thus, Morayne in 1987 considered the
following question: can function $F=\la f_1,f_2\ra$ be chosen in such a way that 
at every point $x\in\real$ either $f_1$ or $f_2$ is differentiable? The
surprising answer is given below. 

\medskip
\begin{theorem} [Morayne~\cite{Morayne}]%3.1
\label{th:Morayne}
  The Continuum Hypothesis is
equivalent to the existence  of a function $F=\la f_1,f_2\ra$ from $\real$ onto
$\real^2$ such that  at every point $x\in\real$ either $f_1$ or $f_2$ is
differentiable. 
\end{theorem}

\medskip
The proof of this theorem is based on a well known theorem of Sierpi\'nski 
\cite[Property $P_1$]{Si:CH} from~1919 that CH is equivalent to the existence of
a decomposition of $\real^2$ into two sets $A$ and $B$ such that  all horizontal
sections of $A$ and all vertical sections of $B$ are at most countable.  It is
also worthwhile to point out that the function $F$ from Theorem~\ref{th:Morayne}
is not a Peano curve, since  it is not continuous. In fact Morayne proves in the
same paper that for such an $F=\la f_1,f_2\ra$ it is impossible that even one of 
$f_1$ or $f_2$ is measurable. 

\medskip
Next, recall that if two continuous functions $f,g\colon\real\to\real$ agree on
some dense set $M\subset\real$ then they are equal. Does the statement remain
true if the clause ``agree on $M$'' is replaced by ``$f[M]=g[M]$?'' Clearly not,
as shown by $M=\rational$ and any two different rational  translations of the
identity function.  What about finding some more complicated set
$M\subset\real$ for which the implication 
\[
\text{if $f[M]=g[M]$ then $f=g$}
\] 
holds for any continuous $f$ and $g$? Even this is  too much to ask, as
recently noted by Burke and Ciesielski
\cite[Remark 6.6]{BC}. On the other hand, the following theorem of Berarducci
and Dikranjan from~1993 gives a positive (consistent) answer to this question in
the class of continuous nowhere constant functions.  (A function is nowhere
constant if it is not constant on any non-empty open set.)

\medskip
\begin{theorem}[Berarducci, Dikranjan \cite{BD}]%3.2
\label{th:BerDikra}
If the Con\-ti\-nu\-um
Hy\-po\-the\-sis holds then there  exists a set $M\subset\real$ (called {\rm magic})
such that for every continuous nowhere constant functions
$f,g\colon\real\to\real$, 
\[
\text{if $f[M]\subset g[M]$ then $f=g$.}
\] 
\end{theorem}

\medskip
The construction of a magic set given in~\cite{BD} is done by an easy diagonal
transfinite induction argument and uses only the assumption that less than
continuum many meager sets do not cover $\real$. In particular, CH can be
replaced by MA in Theorem~\ref{th:BerDikra}. 

Examining the problem of existence of a magic set in ZFC Burke and Ciesielski
noticed the following properties of a magic set.

\medskip
\begin{theorem}[Burke, Ciesielski \cite{BC}]%3.3
\label{th:CBurke1}
If $M\subset\real$ is a 
magic set then
\begin{itemize}
\item[(a)] $M$ is dense and nowhere meager;
\item[(b)] $f[M]\not\supset[0,1]$ for every continuous $f\colon\R\to\R$.
\end{itemize} 
\end{theorem}

\medskip
In fact part (b) of Theorem~\ref{th:CBurke1} is just a remark: if there were a
continuous $f\colon\R\to\R$ with $f[M]\supset[0,1]$ then it could be easily
modified to  a nowhere constant function such that $f[M]=\real$,  and the
functions $f$ and $g=1+f$  would give a contradiction.  But (b) shows that there
is no magic set of cardinality continuum in the model from
Theorem~\ref{th:sierp2NEG}, the iterated perfect set model. Although  it was
noticed in \cite{BC} that in this model there exists a magic set (clearly of
cardinality less than $\continuum$), Theorem~\ref{th:CBurke1} was used by
Ciesielski and Shelah as a base in proving that magic set cannot be constructed
in ZFC.

\medskip
\begin{theorem}[Ciesielski, Shelah \cite{CShel}]%3.4
\label{th:CShelah}
There is a model of ZFC in which 
\begin{itemize}
\item[(a)] every subset of\/ $\R$ of cardinality less than $\continuum$ is
           meager;
\item[(b)] for every set $M\subset\R$ of cardinality continuum there exists
           a continuous function $f\colon\R\to\R$ such that $f[M]=[0,1]$.
\end{itemize} 
In particular, there is no magic set in this model. 
\end{theorem}

\medskip
The magic sets for different classes of functions have also been considered.
Burke and Ciesielski~\cite{BC}  studied  such sets (which they call {\em sets of
range uniqueness}\/)  for the classes of measurable functions with respect to
abstract measurable spaces with negligibles. In particular, they proved the
following theorem  concerning the Lebesgue measurable functions.

\newpage
%\medskip
\begin{theorem}[Burke, Ciesielski \cite{BC}]%3.5
\label{th:CBurke2}
\
\begin{itemize}
\item[(a)] If $\cov(\calN)=\continuum$ (thus under CH or MA) then 
           there exists a set $M\subset\R$ with the property that 
           for every measurable functions $f,g\colon\R\to\R$
           which are not constant on any set of positive measure
           \[
           \text{if $f[M]\subset g[M]$ then $f=g$ almost everywhere.}
           \]
\item[(b)] There is a model of ZFC in which a set from part (a) does not exist. 
\end{itemize} 
\end{theorem}

\medskip
The model satisfying Theorem~\ref{th:CBurke2}(b)  is a modification 
of the iterated
perfect set model and  was constructed by Corazza \cite{Corazza} in~1989.  Once
again it satisfies property (b) of Theorem~\ref{th:CShelah},  while part (a) is
replaced by $\cov(\calN)=\continuum$. It has also been proved by 
Burke, Ciesielski, and 
Larson that for the class $D^1$ of differentiable functions the
existence of a magic set can be proved in ZFC.

\medskip
\begin{theorem}[Burke, Ciesielski \cite{CL:magic}]%3.6
\label{th:CLmagic}
There exists a set 
$M\subset\R$ such that for every $D^1$  nowhere constant functions
$f,g\colon\R\to\R$
           \[
           \text{if $f[M]\subset g[M]$ then $f=g$.}
           \] 
\end{theorem}

\medskip \noindent
Note also that the existence of a countable magic set 
(a convergent sequence) for the class
of analytic functions has been proved already in~1981 
by Diamond, Pomerance, and Rubel~\cite{DPR}.
However, not all convergent sequences form a magic set for this class.

\medskip
For the following consideration recall that a function $f\colon\R^n\to\R$ is
{\em Darboux} (or {\em has the Darboux property}\/) if $f[C]$ is connected for
every connected  subset $C$ of $\R^n$. Thus, in case of $n=1$ Darboux functions
are precisely the functions for which the Intermediate Value Theorem holds. The
class of Darboux functions will be denoted here by $\D$ (with $n$ clear from the
context, usually $n=1$). 

The class of Darboux functions has been studied for a long time as one of
possible generalizations of the class of continuous functions.  (Clearly every
continuous function is Darboux.) However, it has some peculiar properties. For
example, it is not closed under  addition. In fact, in 1927 Lindenbaum \cite{Li}
noticed (without a proof)  that every function $f\colon\real\to\real$ can be
written as a sum of two Darboux functions. (For proofs, see \cite{Si3, SMa}.) 
This theorem has been improved in several ways.  Erd\H{o}s~\cite{Erdos} showed
that if $f$ is measurable, both of the summands can be chosen to be measurable. 
Another improvement was done by Fast~\cite{F} in~1959 who proved that for every
family 
$\F$ of real functions that has cardinality  continuum  there is just one 
Darboux function
$g$ such that the sum of $g$ with any function in $\calF$ has the Darboux
property.  The natural question  of whether such a ``universal'' summand exists
also for families of larger cardinality has been studied by Natkaniec~\cite{N1} 
and lead to the development described in Section~4. %\ref{sec:CardF}. 

A problem that is in some sense opposite to the existence of a ``universal'' 
summand is for which families $\calF$ of functions there is a ``universally
bad''  Darboux function $g$, in the sense that the sum of $g$ with any function
in
${\calF}$ does not have the Darboux property.  In~1990 Kirchheim and Natkaniec
addressed this problem for the class $\F$  of continuous nowhere constant
functions.

\medskip
\begin{theorem}[Kirchheim, Natkaniec~\cite{NK}] %3.7
\label{th:NatKir}
 If union of less than
$\continuum$ many meager subsets of\/ $\real$ is meager (thus under CH or MA)
then there exists a Darboux function $g\colon\R\to\R$ such that $f+g$ is not
Darboux for every  continuous nowhere constant function $f\colon\R\to\R$.
\end{theorem}

\medskip
The problem whether the additional set-theoretic assumptions are necessary in
this theorem was investigated in~1992  by Komj\'ath~\cite{Km} and was settled
in~1995 by Stepr\=ans.

\medskip
\begin{theorem}[Stepr\=ans~\cite{St}]%3.8
\label{th:Stepr1}
 It is consistent with ZFC that
for every  Darboux function $g\colon\R\to\R$ there exists  a continuous nowhere
constant function $f\colon\R\to\R$ such that $f+g$ is Darboux.
\end{theorem}

\medskip
A model having this property is the iterated perfect set model. 
Note also that in Theorem~\ref{th:NatKir} the restriction to
the nowhere constant functions is important.
This has been proved independently by T.~Natkaniec  
(in his 1992$/$93 paper~\cite{Na2a})
and by J.~Stepr\=ans (in the~1995 paper mentioned above). 

\medskip
\begin{theorem}[Natkaniec~\cite{Na2a}, Stepr\=ans~\cite{St}]%3.9
\label{th:Stepr2}  
For every Dar\-bo\-ux \linebreak func\-tion
$g\colon\R\to\R$ there exists  a continuous non-constant function
$f\colon\R\to\R$ such that $f+g$ is Darboux.
\end{theorem}

\medskip
To state further results recall the following generalizations of continuity. A
function $f\colon\R^n\to\R$ is {\em almost continuous} (in the sense of Stallings) 
if each open subset of $\real^n\times\real$ containing the graph of
$f$ contains also a continuous function from $\R^n$ to $\R$~\cite{Stal}.
Function $f\colon\R\to\R$ {\it has a perfect road\/} at
$x\in\R$ if there exists a perfect set $C$ such that $x$ is a bilateral limit
point of $C$ and $f\restriction C$ is continuous at $x$~\cite{Max}. The classes
of all almost continuous functions and all functions having a perfect road at
each point are denoted by $\op{AC}$ and $\op{PR}$, respectively.  It is easy to
see that $\C\subset\op{AC}\subset\D$ (for functions on $\real$) and that the
inclusions are strict (see e.g.~\cite{BHL}), where $\C$ stands for the class of
all continuous functions.  We will also consider the class $\op{SZ}$ of  {\em
Sierpi\'nski-Zygmund (SZ-) functions}, i.e., functions $f\colon\real\to\real$
whose restrictions $f\restriction X$ are discontinuous for all subsets $X$ of
$\R$ of cardinality continuum.  (That is, functions from Theorem~\ref{th:SZ}.)

The classes $\op{SZ}$ and $\op{PR}$ recently appeared in a~1993 paper of Darji
\cite{D}, who constructed in ZFC a function $f\in\op{SZ}\cap\op{PR}$.  Answering
a question posed by Darji the following theorem has been proved
recently by Balcerzak, Ciesielski, and Natkaniec.

\medskip
\begin{theorem}[Balcerzak, Ciesielski, Natkaniec \cite{BCN}]%3.10
\label{th:BCN}
\
\begin{itemize}
\item[(a)] If\/ $\R$ is not a union of less
           than continuum many of its meager subsets (thus under CH or MA)
           then there exists an $f\in\op{SZ}\cap\op{PR}\cap\op{AC}$.
\item[(b)] There is a model of ZFC in which every
           Darboux function $f\colon\real\to\real$
           is continuous on some set of cardinality continuum. 
           In particular, in this model we have 
           $\op{SZ}\cap\op{AC}=\op{SZ}\cap\D=\emptyset$. 
\end{itemize} 
\end{theorem}

\medskip
The model satisfying Theorem~\ref{th:BCN}(b) is, once again, the iterated
perfect set model. 

\medskip
Another generalization of continuity is that of countable  continuity: a
function $f\colon\R\to\R$ is  {\em is countably continuous} if there exists a
countable partition  $\left\{X_n\right\}_{n=1}^\infty$ of $\R $   such that the
restriction of $f$ to any $X_n$ is continuous. (See also
Section~4.) %\ref{sec:CardF}.) 
In~1995 Darji gave the following combinatorial
characterization of this notion. 

\medskip
\begin{theorem}[Darji \cite{D1,D2}]%3.11
\label{th:DarjCountCon}
 If the Con\-ti\-nu\-um Hy\-po\-the\-sis holds \linebreak then
\begin{itemize}
\item[($\star$)] 
$f\colon\R\to\R$ is countably continuous  if and only if for every uncountable
set $U\subseteq\R $  there is an uncountable set $V\subseteq U$ such that the
restriction $f\restriction V$ is continuous.
\end{itemize} 
\end{theorem}

\medskip
The characterization ($\star$) cannot be proved in ZFC. This follows from a
result of Cicho\'n and Morayne~\cite{CiMo} from 1988 which  implies that in some
models of ZFC  (actually, when 
$\continuum=\omega_2$ and $d=\omega_1$, where $d$ is the dominating number)  
($\star$) is false.  However, it is not known, whether the equivalence ($\star$)
can be proved in absence of CH, leading to the following open problem.

\medskip
\begin{problem}%4
\label{pr:Darj}
{\em Is ($\star$) from Theorem~\ref{th:DarjCountCon} equivalent to the
Continuum Hypothesis?}
\end{problem}

\medskip
Another recent theorem concerning countable and symmetric continuities is the
following theorem of Ciesielski and Szyszkowski, answering a question of
L.~Larson. 

\medskip
\begin{theorem}[Ciesielski, Szyszkowski \cite{CSz}]%3.12
\label{th:CSz} 
There exists a
sym\-me\-tri\-ca\-lly continuous function $f\colon\R\to\R$  such that for some set
$Z\subset\real$ of cardinality continuum 
$f\restriction Z$ is of Sierpi\'nski-Zygmund type, i.e., 
$f\restriction X$ is discontinuous for any subset $X$ of $Z$ of cardinality
continuum.

In particular, $f$ is not countably continuous. 
\end{theorem}

\medskip
We will finish this section with the following  two interesting results. The
first one was proved independently in~1978 by  Grande and Lipi\'nski and
in~1979 by Kharazishvili.

\medskip
\begin{theorem}[Grande,\  Lipi\'nski~\cite{GrLp},\  Kharazishvili~\cite{Kh}] 
%3.13
\label{th:GLandK}
\ If \  the \linebreak
Con\-ti\-nu\-um Hypothesis holds then  there exists a non-measurable function 
$F\colon\R^2\rightarrow\R$ such that for every measurable 
$f\colon\R \rightarrow\R$,  the composition $F(x,f(x))$ is measurable. 
\end{theorem}

\medskip
This theorem has important consequences concerning the existence of solutions of
the differential equation $y^{\prime }=F(x,y)$ in the class of absolutely
continuous functions.  In~1992 Balcerzak~\cite{B1} showed that in
Theorem~\ref{th:GLandK} the CH assumption can be weakened to
$\non(\N)=\continuum$. However, the following problem remains open.

\medskip
\begin{problem}%5
\label{Balc}
{\em Can Theorem~\ref{th:GLandK} be proved in ZFC?}
\end{problem}

\medskip \noindent
In fact, all functions $F$ satysfying Theorem~\ref{th:GLandK}
are of the form $\charf{h}$, where $h$ is a (partial) function from $\real $
to $\real $.
It is worth to mention here that, by~\cite[Prop.~1.5]{B1}, the property
considered in Problem~\ref{pr:Blum} implies that no $F=\charf{h}$ with
$h\colon\real\to\real$ can satisfy Theorem~\ref{th:GLandK}.
Similarly, the property considered in the following stronger version
of Problem~\ref{pr:Blum} implies that $F$ from Theorem~\ref{th:GLandK}
cannot be of the form $\charf{h}$ for a function $h$ from $Y\subset\real$
into $\real$.

\medskip
\begin{problem}%6
\label{BalcGen}
Is it consistent that for every subset $Y$ of $\real$ 
of positive outer measure and every function $f\colon Y\to\real$ 
there exists a set $X\subset Y$ of positive outer measure
such that $f\restriction Y$ is continuous?
\end{problem}

\medskip
The second result is the following 1974 theorem of R.~O.~Davies. 

\medskip
\begin{theorem}[Davies~\cite{Davies}]%3.14 
\label{th:Davies}
If the Con\-ti\-nu\-um Hypothesis holds
then for every $f\colon\R^2\to\R$ there exist functions $g_n,h_n\colon\R\to\R$,
$n<\omega$, such that
\[ f(x,y)=\sum_{n=0}^\infty g_n(x)\cdot h_n(y).
\] 
\end{theorem}

\medskip
\begin{problem}[{\cite[Problem~15.11]{millerq}}]%7
\label{pr:Davies}
{\em Is the con\-clu\-sion of
Theorem~\ref{th:Davies} \linebreak e\-qu\-i\-va\-lent to CH?}
\end{problem}

\medskip
Note that Theorem~\ref{th:Davies} is related to Hilbert's Problem~13
(from his famous Paris lecture of~1900) 
and a~1957 theorem of Kolmogorov, in which he proves 
that every continuous function $f\colon[0,1]^n\to\real$ 
can be represented in a certain form (similar to the above) by 
continuous functions of one variable. 
An interesting account on this and related results 
can be found in a~1984 paper of 
Sprecher~\cite{Sprec}.

%\input{survey2}

\medskip
\stepcounter{section}
\section*{\bf 4. Cardinal functions in analysis} 
\label{sec:CardF}

\medskip
The important recent developments in set theoretical analysis concern the
cardinal functions that are defined for different classes  of real functions.
These investigations seem to be analogous to 
those concerning of cardinal functions in topology from the 1970's and 1980's.
(See \cite{Ju2,Ho,Ju3,V}.) 
They are also related to the deep studies of cardinal invariants 
associated with different small subsets of the real line. 
(For a summary of the results concerning cardinals related to the measure
and category see~\cite{Fr} or \cite{BJ}. 
For a survey concerning cardinals associated with the 
thin sets derived from harmonic analysis
see~\cite{BKR}.)

The first group of functions is motivated by the notion of countable continuity
and was introduced in~1991  by J.~Cicho\'{n}, M.~Morayne, J.~Pawlikowski, and
S.~Solecki in~\cite{cimopaso}.  More precisely, they define  the {\em
decomposition} function
$\dec(\F,\G)$ for arbitrary  families $\F\subset\real^\real$ and
$\G\subset\bigcup\{\real^X\colon X\subset\real\}$, where $Y^X$ stands for the
set of all functions from $X$ to $Y$.
\[
\dec(\F,\G)=\min\{\kappa\leq\co\colon(\forall f\in\F) (\exists {\cal
X}\in\Pi_\kappa) (\forall X\in{\cal X})(f\restriction X\in\G)\}\cup\{\co^+\},
\]
where $\Pi_\kappa$ denotes the family of all coverings of $\real$ with at
most $\kappa$ many sets. In particular, if $\C$ stands for the family of all
continuous functions (from subsets of $\real$ into $\real$) then
\[
\text{$f\colon\real\to\real$ is countably continuous if and only if
$\dec(\{f\},\C)\leq\omega$.}
\] 
In~\cite{cimopaso} the authors considered the values  of
$\dec(\B_\beta,\B_\alpha)$ for $\alpha<\beta<\omega_1$, where
$\B_\alpha$ stands for the functions of $\alpha$-th Baire class. 

The motivation for this definition comes from a question of N.~N.~Luzin
whether every Borel function is countable continuous. 
This question was answered negatively by P.~S.~Novikov 
(see  Keldy\v{s}~\cite{Keld})
and was subsequently generalized by Keldy\v{s}~\cite{Keld} 
(in~1934), and S.~I.~Adian and P.~S.~Novikov~\cite{AdiNov} (in~1958). 
The most general result in this direction was obtained in late 1980's
by M.~Laczkovich 
(see Cicho\'{n}, Morayne~\cite{CiMo}) who proved, in particular, that 
$\dec(\B_\beta,\B_\alpha)>\omega$ for every $\alpha<\beta<\omega_1$.

One of the most interesting results from the paper~\cite{cimopaso} 
is the following theorem. 

\medskip
\begin{theorem}[Cicho\'{n}, Morayne, Pawlikowski, Solecki~\cite{cimopaso}]%4.1
\label{th:CMPS}
\[ {\rm cov}(\M)\leq \dec(\B_1,\C)\leq d,
\] where ${\rm cov}(\M)$ is the smallest cardinality of a covering of $\real$ by 
meager sets, and $d$, the dominating number, is the smallest cardinality of a
dominating family $D\subset\omega^\omega$, i.e., such that for every
$f\in\omega^\omega$ there exists $g\in D$ with
$f\leq^* g$.%
\footnote{Recall that for $f,g\in\omega^\omega$ we write $f\leq^* g$
if there exists an $n<\omega$ such that $f(m)\leq g(m)$ for every $m>n$.}
\end{theorem}

\medskip \noindent 
It has been also shown by J.~Stepr\={a}ns and  S.~Shelah that
none of these inequalities can be replaced by the equation.

\medskip
\begin{theorem}[Stepr\={a}ns~\cite{step.30}]%4.2
\label{th:Stdec}
It is consistent with ZFC
that 
\[ {\rm cov}(\M)<\dec(\B_1,\C).
\] 
\end{theorem}

\medskip
\begin{theorem}[Shelah, Stepr\={a}ns~\cite{step.34}]%4.3
\label{th:ShStdec}
 It is consistent with ZFC that 
\[
\dec(\B_1,\C)<d.
\]
\end{theorem}

\medskip
There are also some interesting results concerning the value of $\dec(\C,\D^1)$,
where $\D^1$ is the class of all (partial) differentiable functions. It has been
proved by Morayne
(see Stepr\={a}ns~\cite[Thm~6.1]{step.38}) that

\medskip
\begin{theorem}[{Morayne~\cite[Thm~6.1]{step.38}}]%4.4
\label{th:Mordec}
$ \cov (\M)\leq\dec(\C,\D)\leq\co$.
\end{theorem}

\medskip
Also, Stepr\={a}ns proved  that 

\medskip
\begin{theorem} [Stepr\={a}ns~\cite{step.38}]%4.5
\label{th:StepDiff}
It is consistent with ZFC that 
$$ 
\dec(\C,\D)<\co.
$$
\end{theorem}

\medskip \noindent
However, the relation between numbers $\dec(\C,\D)$, 
$\dec(\B_1,\C)$ and 
$\dec(\B_\beta,\B_\alpha)$ for $0<\alpha<\beta<\omega_1$ is unclear. 

In the same direction, K.~Ciesielski recently noticed that (obviously)
\[
\cf(\co)\leq\dec(\op{SZ},\C)\leq\co
\] 
and that it is the best that can be said in ZFC.

\newpage
%\medskip
\begin{theorem}[Ciesielski \cite{KCnowa}] %4.6
\label{th:KCSZdec}
\
\begin{itemize}
\item[(1)] For every $\kappa$ with $\cf(\kappa)>\omega$ there exists a model of
ZFC in which $\co=\kappa$ and
$\dec(\op{SZ},\C)=\co$.
\item[(2)] For every $\kappa$ with $\cf(\kappa)>\omega$ there exists a model of
ZFC in which $\co=\kappa$ and
$\dec(\op{SZ},\C)=\cf(\kappa)=\cf(\co)$.
\end{itemize} 
\end{theorem}

\medskip
In fact, (1) happens in a model obtained by extending a ground model with  GCH
by adding $\kappa$ many Cohen reals. The equation $\dec(\op{SZ},\C)=\co$ follows
immediately from Theorem~\ref{th:Sh1}.

The model for (2) is obtained as follows.  
You start with a model with GCH,  
assume that $\lambda={\rm cf}(\kappa)<\kappa$
and take an increasing sequence \linebreak
$\{\lambda_\xi\colon\xi<\lambda\}$
cofinal with $\lambda$ and such that each $\lambda_\xi$ is a cardinal successor. 
The desired model is obtained by a generic extension
via forcing $P$ which a finite support iteration of forcings 
$M_\xi$, where each $M_\xi$ is a standard ccc forcing
adding the Martin's Axiom over the previous model and making
$\co=\lambda_\xi$.

\medskip
The second group of cardinal functions is defined  in terms of algebraic
operations on functions.  Their definition  was motivated by the following
property of Darboux functions (from $\R$ to $\R$) due to Fast and mentioned in
the previous section:
\begin{itemize}
\item for every family ${\cal H}\subset\real^\real$ with
$|{\cal H}|\leq\continuum$ 
\begin{equation}
\label{con1}
\text{ there exists $g\in\real^\real$ such that
$g+h\in\D$ for every $h\in {\cal H}$, }
\end{equation}
\end{itemize} 
where $|Z|$ denotes the cardinality of $Z$.  In~1974
Kellum~\cite{Ke}  proved the similar result for the class $\op{AC}$ of almost
continuous functions and in~1991 Natkaniec~\cite{N1} defined the following
cardinal functions for every ${\calF}\subseteq\R^{\R}$ to study these phenomena
more closely. 
\begin{align*}
\op{A}({\calF}) &= %& 
\min\left\{\left|{\cal H}\right|\colon{\cal H}\subset\real^\real\ \text {\&}\ 
\neg\exists g\in\R^{\R }\ 
\forall h\in{\mathcal{H}}\ g+h\in{\calF}\right\}
\cup\{(2^\continuum)^+\}\\ 
&= %& 
\min\left\{\left|{\mathcal{H}}\right|\colon{\mathcal{H}}\subset\real^\real\ \text{\&}\ 
\forall g\in\R^{\R }\ 
\exists h\in{\mathcal{H}}\ g+h\notin{\calF}\right\}
\cup\{(2^\continuum)^+\}\\
\op{M}({\calF}) \!\! &= %& 
\!\!
\min\left\{\left|{\mathcal{H}}\right|\colon{\mathcal{H}}\subset\real^\real\ \text{\&}\ 
\neg\exists g\in\R^{\R}\setminus\{\charf\emptyset\}\ 
\forall h\in{\mathcal{H}}\ g\cdot h\in{\calF}\right\}
\cup\{(2^\continuum)^+\}\\
\!\! & =%& 
\!\!
\min\left\{\left|{\mathcal{H}}\right|\colon{\mathcal{H}}\subset\real^\real\ \text{\&}\ 
\forall g\in\R^{\R}\setminus\{\charf\emptyset\}\ 
\exists h\in{\mathcal{H}}\ g\cdot h\notin{\calF}\right\}
\cup\{(2^\continuum)^+\}.
\end{align*}\label{defMUL}
The extra assumption that 
$g\neq\charf\emptyset$ is added in the definition of $\op{M}$ since  otherwise
for every family $\F\su\real^\real$ containing constant zero function
$\charf\emptyset$ we would have $\op{M}(\F)=(2^\co)^+$.

Its easy to see that the functions $\op{A}$ and $\op{M}$ are monotone in a sense
that 
$\op{A}(\F)\leq\op{A}(\G)$ and $\op{M}(\F)\leq\op{M}(\G)$ for every
$\F\subset\G\subset\R^\R$.  Also clearly (\ref{con1}) is false for
${\mathcal{H}}=\real^\real$. Thus, in language of the function $\op A$
the results of Fast and Kellum can be  expressed as follows:
\[
\continuum<\op{A}(\op{AC})\leq\op{A}(\D)\leq 2^\continuum.
\] 
If $2^\continuum=\continuum^+$ (so, under the Generalized Continuum
Hypothesis GCH) the values of $\op{A}(\op{AC})$  and $\op{A}(\D)$ are clear:
$\op{A}(\op{AC})=\op{A}(\D)=2^\continuum=\continuum^+$.  Thus, Natkaniec
asked~\cite[p. 495]{N1} (see also \cite[Problem 1]{GMN}) whether the equation
$\op{A}(\op{AC})=2^\continuum$ can be proved in ZFC.  

This question was investigated by Ciesielski and Miller in~1994. They proved
that $\op{A}(\op{AC})=\op{A}(\D)$, that  the cofinality $\op{cf}(\op{A}(\D))$ of
$\op{A}(\D)$ is greater than $\continuum$, and that this together with the
inequalities 
$\continuum<\op{A}(\D)\leq 2^\continuum$ is essentially all  that can be proved
in ZFC.

\smallskip
\begin{theorem}[Ciesielski, Miller \cite{CMi}]%4.7
\label{th:CMill1}
\
\begin{itemize}
\item[(a)] $\op{A}(\op{AC})=\op{A}(\D)={\frak e}_\continuum$, where 
\[ {\frak e}_\kappa=\min\{|F|\colon F\su \kappa^\kappa\ \text{\em \&}\ \forall
                g\in \kappa^\kappa\ \exists f\in F\ |f\cap g|<\kappa\}.
\]
\item[(b)] $\op{cf}(\op{A}(\D))>\continuum$.
\item[(c)]
  Let $\lambda\geq\kappa\geq\omega_2$ be cardinals such that
  $\op{cf}(\lambda)>\omega_1$ and
  $\kappa$ is regular.  Then it is relatively consistent with
  ZFC that the Continuum Hypothesis is true,
  $2^{\continuum}=\lambda$, and $\op{A}(\D)=\kappa$.
\item[(d)]
  Let $\lambda$ be a cardinal such that
  $\op{cf}(\lambda)>\omega_1$.
  Then it is relatively consistent with
  ZFC that the Continuum Hypothesis holds and
 $\op{A}(\D)=\lambda=2^{\continuum}$.
\end{itemize}  
\end{theorem}

\smallskip \noindent
In particular Theorem~\ref{th:CMill1} says that 
$\op{A}(\D)$ does not have to be a regular cardinal (part (d)) and that
$\op{A}(\D)$  can be any regular cardinal number  between
$c^+$ and $2^c$, with $2^{c}$ being ``arbitrarily large'' (part (c)).

At the same time Natkaniec and Rec\l aw established the values of
$\op{M}(\op{AC})$ and $\op{M}(\D)$ proving

%\smallskip
\begin{theorem} [Natkaniec, Rec\l aw \cite{NR}]%4.8
\label{th:NR} 
$\op{M}(\op{AC})=\op{M}(\D)=\cf(\co)$.
\end{theorem}

\smallskip
The first systematic study of functions $\op{A}$ and $\op{M}$  was done by
Ciesielski and Rec\l aw in the later part of~1995. They collected basic
properties of operators $\op{A}$ and $\op{M}$, which are stated below, and 
found the values of $\op{A}$ and $\op{M}$ for some other classes of functions.

%\smallskip
\begin{proposition}[\cite{CR}] %4.9
\label{prop:CRec} Let $\emptyset\neq\F\su\G\su\reals^\reals$.
Then
\begin{itemize}
\item[(0)] $\add(\emptyset)=\mul(\emptyset)=1$; 
\item[(1)] $\add(\F)\leq\add(\G)$ and $\mul(\F)\le\mul(\G)$;
\item[(2)] $\add(\F)\geq 2$;
\item[(2$^\prime$)] $\mul(\F)\geq 2$ if $\charf\emptyset,\charf\real\in\F$;
\item[(3)] $\add(\F)=\left(2^\co\right)^+$ if and only if $\F=\real^\real$;
\item[(3$^\prime$)] $\mul(\F)\leq\co$ if $r\charf{\{x\}}\not\in\F$ for every
           $r,x\in\real$, $r\neq 0$;
\item[(4)] $\add(\F)=2$ if and only if
$\F-\F=\{f_1-f_2\colon f_1,f_2\in\F\}\neq\real^\real$.%
\footnote{In~\cite{CR} this was proved with the additional assumption that
$\charf\emptyset\in\F$. This extra assumption was removed by F.~Jordan
in~\cite{FJordan}.}
\end{itemize} 
\end{proposition}

\medskip \noindent
 In particular, (4) from Proposition~\ref{prop:CRec} shows
that every function is a difference of two functions from a class $\F$ if and
only if $\add(\F)>2$. 

To state the other results from~\cite{CR} recall the definitions the following
classes of functions, where $X$ is an arbitrary topological space. 

\begin{itemize}
\item[$\conn(X)$] of {\em connectivity functions} $f\colon X\to\R$, i.e., such
          that the graph of $f$ restricted to $C$ (that is $f\cap[C\times\R]$)
          is connected in $X\times\R$ for every connected subset $C$ of $X$.
\item[$\ext(X)$] of {\em extendable functions} $f\colon X\to\R$, i.e., such that
          there exists a connectivity function $g\colon X\times[0,1]\to\R$ 
          with $f(x)=g(x,0)$ for every $x\in X$.
\item[$\pc(X)$] of {\em peripherally continuous functions} $f\colon X\to\R$, 
          i.e., such that
          for every $x\in X$ and any pair $U\su X$ and $V\in\R$ of 
          open neighborhoods of $x$ and $f(x)$, respectively, 
          there exists an open neighborhood $W$ of $x$ with 
          $\cl(W)\su U$ and $f[\bd(W)]\su V$, where $\cl(W)$
          and $\bd(W)$ stand for the closure and the boundary of $W$,
          respectively.
\end{itemize} 
We will write 
$\conn$, $\ext$ and $\pc$ in place of $\conn(X)$, $\ext(X)$, and $\pc(X)$ if
$X=\real$.  Notice also, that $f\in\pc$ if and only if $f$ is weakly continuous,
as defined on page~\pageref{pageWC}.

For the generalized continuity classes of functions  (from $\real$ into $\real$)
defined so far  we have the following proper inclusions $\subset$, marked by
arrows
$\longrightarrow$. (See~\cite{BHL}.)

\begin{picture}(300,100)
\thicklines
\put(10,40){\makebox(0,0){$\ext$}}
\put(80,80){\makebox(0,0){$\op{AC}$}}
\put(160,15){\makebox(0,0){$\op{PR}$} }
\put(160,80){\makebox(0,0){$\conn$}}
\put(240,80){\makebox(0,0){$\D$} }
\put(320,40){\makebox(0,0){$\pc$}}

\put(30,50){\vector(1,1){30}}
\put(100,80){\vector(1,0){30}}
\put(190,80){\vector(1,0){30}}
\put(260,80){\vector(3,-2){40}}
\put(30,35){\vector(4,-1){100}}
\put(190,10){\vector(4,1){100}}

\end{picture}
 
\centerline{Chart 1.}

\medskip
In particular, inclusions $\op{AC}\subset\conn\subset\D$, monotonicity of $\op
A$ and Theorem~\ref{th:CMill1}(a) imply that
$\add(\conn)=\add(\op{AC})=\add(\D)$. Similarly, Theorem~\ref{th:NR} implies
that $\mul(\conn)=\mul(\op{AC})=\mul(\D)$. The values of $\op A$ and $\op M$ for
the remaining classes  are as follows. 

%\newpage
\medskip
\begin{theorem}[Ciesielski, Rec\l aw \cite{CR}]%4.10
\label{th:CRec}
\
\begin{itemize}
\item[(1)] $\add(\ext)=\add(\op{PR})=\co^+$.
\item[(2)] $\add(\pc)=2^\co$.
\item[(3)] $\mul(\ext)=\mul(\op{PR})=2$.
\item[(4)] $\mul(\pc)=\co$.
\end{itemize} 
\end{theorem}

\medskip \noindent
 Notice also that 
$\ext\subset\op{AC}\cap\op{PR}\subset\op{Conn}\cap\op{PR}
\subset{\mathcal{D}}\cap\op{PR}\subset\op{PR}$. Thus, by monotonicity of $\op A$
and the above theorem we obtain the following corollary. 

\begin{corollary}%4.11
\label{cor:KCcap}
\ 
\begin{itemize}
\item[(1)] $\add(\op{AC}\cap\op{PR})=\add(\op{Conn}\cap\op{PR})=
\add({\mathcal{D}}\cap\op{PR})=\co^+$; and,
\item[(2)] 
$\mul(\op{AC}\cap\op{PR})=\mul(\op{Conn}\cap\op{PR})=
\mul({\mathcal{D}}\cap\op{PR})=2$.
\end{itemize} 
\end{corollary}

\medskip
The values of functions $\add$ and $\mul$ for the class $\op{SZ}$ has been
studied by Ciesielski and Natkaniec. First they noticed that 
if the definition of $\mul$ from page~\pageref{defMUL} is used
then trivially $\mul(\op{SZ})=1$, since for any function 
$h\in\real^\real$ with $\left|h^{-1}(0)\right|=\co$
we have $g\cdot h\notin\op{SZ}$ for every $g\in\real^\real$. 
Thus, they modified the definition of $\mul(\op{SZ})$ to
\begin{align*}
\op{M}(\op{SZ}) & = %& \!\!
\min\left\{\left|{\mathcal{H}}\right|\colon{\mathcal{H}}\subseteq {\mathcal{R}}_0\ 
\text{\&}\ 
\neg\exists g\in\R^{\R}\ 
\forall h\in{\mathcal{H}}\ g\cdot h\in\op{SZ}\right\}
\cup\{(2^\continuum)^+\},
\end{align*}
where 
\[
{\mathcal{R}}_0=\left\{f\in\R^\R\colon \left|f^{-1}(0)\right|<\co\right\}.
\]
With this agreement in place they proved the following result. 

\medskip
\begin{theorem}[Ciesielski, Natkaniec~\cite{CNat}]%4.12
\label{th:CNat1}
\ 
\begin{itemize} 
\item[(a)] $\mul(\op{SZ})=\add(\op{SZ})={\op d}_\continuum$, where 
\[ {\op d}_\kappa=\min\{|F|\colon F\su \kappa^\kappa \&\ \forall
                g\in \kappa^\kappa\ \exists f\in F\ |f\cap g|=\kappa\}.
\]
\item[(b)] 
  Let $\lambda\geq\kappa\geq\omega_2$ be cardinals such that
  $\op{cf}(\lambda)>\omega_1$ and
  $\kappa$ is regular.  Then it is relatively consistent with
  ZFC that the Continuum Hypothesis is true,
  $2^{\continuum}=\lambda$, and $\add(\op{SZ})=\op{A}(\D)=\kappa$.
\item[(c)]
  Let $\lambda>\omega_2$ be a cardinal such that
  $\op{cf}(\lambda)>\omega_1$.
  Then it is relatively consistent with
  ZFC that the Continuum Hypothesis holds, $2^{\continuum}=\lambda$, 
  and $\add(\op{SZ})=\co^+<2^{\continuum}=\op{A}(\D)$.
\end{itemize}  
\end{theorem}

\medskip
However, the following problems remain open.

\medskip
\begin{problem}[{\cite[Problems 2.13 and 2.17]{CNat}}]%8
\label{pr:SZ}
\
\begin{itemize}
\item[(a)] Is it consistent that $\add(\op{SZ})>\op{A}(\D)$?
\item[(b)] Can $\add(\op{SZ})$ be a singular cardinal?
\end{itemize} 
\end{problem}

\medskip
Another systematic study of 
the operator $\op A$ was done by F.~Jordan in~1996. In
his study he examined the values of $\op{A}(\neg\F)$ where
$\neg\F=\R^\R\setminus\F$ and classes $\F$ are chosen from  those discussed
above. Notice that $\op{A}(\neg\F)$ has  the following very nice 
interpretation:
\newpage
\[
\text{$\op{A}(\neg\F)$ is the smallest cardinality of an
${\cal H}\subset\R^\R$ such that $\F-{\cal H}=\R^\R$,}
\] 
where $\F-{\cal H}=\{f-h\colon f\in\F\ \&\ h\in{\cal H}\}$. To make this
study non-trivial Jordan notes first that the value of $\op{A}(\F)$ does not
determine the value of $\op{A}(\neg\F)$:

\medskip
\begin{theorem}[Jordan~\cite{FJordan}]%4.13
\label{th:Jor1}
For every cardinal number 
$2\leq\lambda\leq 2^{\cuum}$ there exists 
$\F\subseteq\real^{\real}$ such that $\add(\F)=2$ and $\add(\neg\F)=\lambda$.

In particular, there exist families $\G,\F\subseteq\real^{\real}$ such that 
$\add(\F)=\add(\G)$ and $\add(\neg\F)\neq\add(\neg\G)$.
\end{theorem}

\medskip
This paper~\cite{FJordan} contains also the following results,
where for a cardinal number $\kappa$ and 
functions $f,g\colon X\to Y$ we define 
$[X]^\kappa=\{Y\subset X\colon|Y|=\kappa\}$
and $[f=g]=\{x\in X\colon f(x)=g(x)\}$.

\medskip
\begin{theorem}[{Ciesielski \cite[Thm.~7]{FJordan}}]%4.14
\label{thm:three} 
$\add(\neg\op{PC})=\omega_{1}$.
\end{theorem}

\medskip
\begin{theorem}[Jordan~\cite{FJordan}]%4.15
\label{th:Jor2}
\
\begin{itemize}
\item[(1)] $\add(\neg\op{PR})=\add(\neg\ext)=2^{\cuum}$.
\item[(2)] $\add(\op{SZ})=d_{\cuum}\leq\add(\neg\D)
           \leq\add(\neg\conn)\leq\add(\neg\ac)\leq d_{\cuum}^{*}$, where
 \begin{multline*}
  d_{\kappa}^{*}
  =\min\{|F|\colon F\subseteq \kappa^{\kappa}\ \&\ 
  (\forall G \in [\kappa^{\kappa}]^{\kappa})(\exists f\in F)(\forall g\in G)\\
  (|[f=g]|=\kappa)\}.
 \end{multline*} 
\item[(3)] $\add(\ac)=\add(\conn)=\add(\D)=
    {\frak e}_{\cuum}\leq\add(\neg\op{SZ})\leq{\frak e}_{\cuum}^{*}$, where
  \begin{multline*}
  {\frak e}_{\kappa}^{*}
  =\min\{|F|\colon F\subseteq \kappa^{\kappa}\ \&\ 
  (\forall G \in [\kappa^{\kappa}]^{\kappa})(\exists f \in F)(\forall g\in G)\\
  (|[f=g]|<\kappa)\}.
  \end{multline*}
\item[(4)] If $\cuum^{<\cuum}=\cuum$ then
   $\same_{\cuum}=\same_{\cuum}^{*}$%
   \footnote{This part was proved by K.~Ciesielski.}
   and $\diff_{\cuum}=\diff_{\cuum}^{*}$.
   In particular
  \[
  \add(\op{SZ})=d_{\cuum}=\add(\neg\D)
           =\add(\neg\conn)=\add(\neg\ac)=d_{\cuum}^{*}
  \]
  and
  \[
\add(\D)=\add(\conn)=\add(\ac)=\diff_{\cuum}=\add(\neg\op{SZ})=\diff_{\cuum}^{*}. 
\]
\item[(5)]  If $\cuum^{<\cuum}=\cuum$ and $\cuum=\lambda^{+}$ 
  for some cardinal $\lambda$ then 
  $\same_{\cuum}\leq\diff_{\cuum}$. In particular
  \[
  \add(\neg\D)=\add(\neg\ac)=\add(\op{SZ})=\same_{\cuum}\leq
  \diff_{\cuum}=\add(\D)=\add(\ac)=\add(\neg\op{SZ}).
  \] 
\end{itemize} 
\end{theorem}

\medskip
The importance of the extra assumptions in (4) and (5) of  Theorem~\ref{th:Jor2}
is not clear.   In particular, the following problem is still open.

\medskip
\begin{problem}%9
\label{prob:one}
When does either $\same_{\cuum}=\same_{\cuum}^{*}$ or 
$\diff_{\cuum}=\diff_{\cuum}^{*}$ hold?
\end{problem}

\medskip
Note also that (4) and (5) of Theorem~\ref{th:Jor2}, and Theorem~\ref{th:CNat1}
imply immediately the following corollary.

\medskip
\begin{corollary}[Jordan~\cite{FJordan}]%4.16
\label{cor:Jor}
\
\begin{itemize}
\item[(1)]  Let $\lambda\geq\kappa\geq\omega_{2}$ be cardinals such that 
$\cf(\lambda)>\omega_{1}$ and $\kappa$ is regular.  Then it is relatively
consistent  with ZFC+CH that $2^{\cuum}=\lambda$ and 
\[
\add(\neg\D)=\add(\neg\ac)=\add(\op{SZ})=
\add(\D)=\add(\ac)=\add(\neg\op{SZ})=\kappa.
\]
\item[(2)] Let $\lambda>\omega_{2}$ be a cardinal such that
$\cf(\lambda)>\omega_{1}$.  Then it is relatively consistent with ZFC+CH that 
$2^{\cuum}=\lambda$, and 
\[
\add(\neg\D)=\add(\neg\ac)=\add(\op{SZ})=\cuum^{+}<
2^{\cuum}=\add(\D)=\add(\ac)=\add(\neg\op{SZ}).
\]
\end{itemize} 
\end{corollary}

\medskip
Finally, the following three classes of functions have been brought to this
picture. 
\begin{itemize}
\item[$\CIVP$] of functions $f\colon\R\to\R$ having the 
 {\it Cantor Intermediate Value Property}, i.e., such that for every 
 $x,y\in\R$ and for each Cantor set $K$ between $f(x)$ and $f(y)$
 there is a Cantor set $C$ between $x$ and $y$ such that $f[C]\subset K$;
\item[$\SCIVP$]
  of functions $f\colon\R\to\R$ having the 
 {\it Strong Cantor Intermediate Value Property}, i.e., such that for every 
 $x,y\in\R$ and for each Cantor set $K$ between $f(x)$ and $f(y)$
 there is a Cantor set $C$ between $x$ and $y$ such that
 $f[C]\subset K$ and $f\restriction C$ is continuous;
\item[$\WCIVP$] of functions $f\colon\R\to\R$ having the 
 {\it Weak Cantor Intermediate Value Property}, that is, such that for every 
 $x,y\in\R$ with \linebreak $f(x)<f(y)$ there is a Cantor set $C$ between
 $x$ and $y$ such that $f[C]\subset(f(x),f(y))$.
 \end{itemize} They fit Chart 1 in the following way.  (See
Gibson, Natkaniec~\cite{G1}.)
%{\bf (Give better reference.)}

\begin{picture}(0,90)
 \put(20,55){\makebox(0,0){$\C$}}
 \put(35,55){\vector(1,0){20}}
  \put(75,55){\makebox(0,0){$\ext$}}
 \put(88,60){\vector(2,1){18}}
 \put(120,70){\makebox(0,0){${\op{AC}}$}}
 \put(132,70){\vector(1,0){30}}
 \put(180,70){\makebox(0,0){$\conn$}}
 \put(195,70){\vector(1,0){20}}
 \put(224,70){\makebox(0,0){${\D}$}}
 \put(233,65){\vector(2,-1){18}}
 \put(270,55){\makebox(0,0){${\pc}$}}
 \put(127,40){\makebox(0,0){${\SCIVP}$}}
  \put(180,40){\makebox(0,0){${\CIVP}$}}
  \put(225,40){\makebox(0,0){${\op{PR}}$}}
\put(145,40){\vector(1,0){20}}
 \put(195,40){\vector(1,0){20}}
 \put(88,52){\vector(2,-1){18}}
 \put(233,43){\vector(2,1){18}}
 \put(195,38){\vector(1,-1){20}}
   \put(235,15){\makebox(0,0){${\WCIVP}$}}
\end{picture}

\centerline{Chart 2: ``Darboux like'' functions.}

\medskip
Clearly the above inclusions, monotonicity of $\add$ and $\mul$, and
Theorem~\ref{th:CRec} imply immediately:
\[
\add(\SCIVP)=\add(\CIVP)=\co^+\ \ \ \text{ and }\ \ \ 
\mul(\SCIVP)=\mul(\CIVP)=2.
\] The values of functions $\add$ and $\mul$  for the class $\WCIVP$, and  for
the classes formed by the intersections of $\op{SZ}$ with each of the remaining
classes mentioned above  were not studied too carefully so far.  However,
obviously $\op{SZ}\cap\SCIVP=\emptyset$ implying
\[
\add(\op{SZ}\cap\SCIVP)=\mul(\op{SZ}\cap\SCIVP)=1.
\] Also, it follows from Theorem~\ref{th:BCN} that 
\[
\text{it is consistent that
$\add(\op{SZ}\cap\D)=\mul(\op{SZ}\cap\D)=1$,}
\] while also
\[
\text{it is consistent that 
$\add(\op{SZ}\cap\op{PR}\cap\op{AC})\geq 2$.}
\] A stronger version of this last inequality follows also from the following
recent theorem of K.~Banaszewski and Natkaniec.

\medskip
\begin{theorem}[Banaszewski, Natkaniec \cite{KBanNat}]%4.17
\label{th:KBanNat}
\
\begin{itemize}
\item[(1)] $\op{SZ}\cap\CIVP\neq\emptyset$. In particular $\SCIVP\neq\CIVP$.
\item[(2)] If union of less than $\continuum$ many meager subsets of $\real$ is
meager (thus under CH and MA) then 
$\op{AC}\cap\CIVP\cap\op{SZ}\neq\emptyset$.
\end{itemize} 
\end{theorem}

\medskip \noindent
 In particular, 
\[
\text{it is consistent that 
$\add(\op{SZ}\cap\CIVP\cap\op{AC})\geq 2$}
\] and \[
\add(\op{SZ}\cap\CIVP)\geq 2.
\] 
This last inequality has been recently improved by F.~Jordan, who proved the
following. 

\medskip
\begin{theorem}[Jordan~\cite{FjordanPr}]%4.18
\label{th:JordanNew}
$\add(\op{SZ}\cap\neg\D\cap\CIVP)>\co$. In particular
\[
\add(\op{SZ}\cap\neg\D\cap\CIVP)=\add(\F)=\add(\op{PR})=\co^+
\] for every $\F\subset\R^\R$ such that 
$\op{SZ}\cap\neg\D\cap\CIVP\subset\F\subset\op{PR}$. 
\end{theorem}

\medskip \noindent
 This theorem gives the
value of $\add$ for many classes that can be obtained intersecting classes
from Chart~2 and $\op{SZ}$. 

Some of the difficulties of studying operators  $\add$ and $\mul$ 
for the intersections $\op{AC}\cap\SCIVP$ and $\op{AC}\cap\CIVP$ 
is that there is relatively little known about these classes.
For example, although Theorem~\ref{th:KBanNat}(2) implies 
that consistently these classes are different, a ZFC example was unknown
until the following very recent theorem of Ciesielski.

\medskip
\begin{theorem} [Ciesielski~\cite{SomeDarbF}]%4.19
\label{th:KCnew}
There exists an $f\in\op{AC}\cap\CIVP$
which is discontinuous on every perfect set.
In particular, $f\notin\SCIVP$.
\end{theorem}

\medskip \noindent
Also the only example of an $f\in\op{AC}\cap\SCIVP\setminus\ext$
results from the following recent theorem of Rosen.
%(See also \cite{SomeDarbF} for a generalization.) 

\medskip
\begin{theorem}[Rosen~\cite{Rosen}] %4.20
\label{th:ROSENnew}
If the Continuum Hypothesis holds then there exists 
an $f\in\op{AC}\cap\SCIVP\setminus\ext$.
\end{theorem}

\medskip \noindent
In fact, the conclusion of Theorem~\ref{th:ROSENnew}
remains true under the assumption that 
union of less than continuum many meager sets is meager. However, the problem of 
existence of such a function in ZFC remains open. 

%\bibitem{Rosen} H.~Rosen,   {\it
%An almost continuous nonextendable function}, preprint. 
%\bibitem{SomeDarbF} K.~Ciesielski,  {\it Some Darboux-like functions}

Several other operators similar to $\add$ and $\mul$ have also been studied.
Thus, in~1995 Natkaniec~\cite{N3} introduced the following operators connected 
to the composition of functions,  where $\Const$ stands for the family of all
constant functions.
\begin{align*}
\cout(\F) & \! = \!  
\min\! \left\{\left|{\mathcal{H}}\right|\!\colon{\mathcal{H}}\subset\real^\real\ \&\ 
\! \neg\exists g\in\R^{\R}\setminus\! \Const\ 
\forall h\in{\mathcal{H}}\ g\circ h\in{\calF}\right\}
\!\cup\!\{(2^\continuum)^+\}\\
\cin(\F)  &\! = \!
\min \!\left\{\left|{\mathcal{H}}\right|\!\colon{\mathcal{H}}\subset\real^\real\ \&\ 
\! \neg\exists g\in\R^{\R}\setminus\! \Const\ 
\forall h\in{\mathcal{H}}\ h\circ g\in{\calF}\right\}
\!\cup\!\{(2^\continuum)^+\}
\end{align*}  
He proved also the following. 

\medskip
\begin{theorem}[Natkaniec~\cite{N3}]%4.21
\label{th:NatComp}
\
\begin{itemize}
\item[(1)] $\cout(\ext)=\cout(\CIVP)=\cout(\op{PR})=1$.
\item[(2)] $\cout(\ac)=\cout(\conn)=\cout(\D)=\cf(\co)$.
\item[(3)] $\cout(\pc)=\co$.
\item[(4)]  $\cin(\ext)=\cin(\ac)=\cin(\conn)=\cin(\D)=1$.
\item[(5)]  $\cin(\CIVP)=\cin(\op{PR})=\cin(\pc)=\co^+$.
\end{itemize} 
\end{theorem}

\medskip
Similar functions have been also studied by Ciesielski and 
Natkaniec~\cite{CNat}: 
\begin{align*}
 \cout(\op{SZ})
&\! =  \!
 \min(\{\left|{\mathcal{H}}\right|\colon{\mathcal{H}}\subseteq{\mathcal{R}}_{out}\ 
 \&\ \neg\exists
 g\in\R^\R\ \forall h\in\mathcal{H}\ g\circ h\in\op{SZ}\}
\cup\{(2^{\co})^+\})\\
 \cin(\op{SZ})
 & \! =  \!
 \min(\{\left|\mathcal{H}\right|\colon \mathcal{H}\subseteq{\mathcal{R}}_{in}\ \&\
\neg\exists
 g\in\R^\R\ \forall h\in\mathcal{H}\ h\circ g\in\op{SZ}\}\cup\{(2^{\co})^+\})
\end{align*} 
where ${\mathcal{R}}_{out}$ (${\mathcal{R}}_{in}$) is the set of all
$h\in\R^\R$ for which there exists $g\in\R^\R$ such that $g\circ h\in\op{SZ}$
($h\circ g\in\op{SZ}$, respectively). 
In fact, the classes ${\mathcal{R}}_{out}$ and ${\mathcal{R}}_{in}$ have the
following nice characterizations:
\[
{\mathcal{R}}_{out}=\left\{f\in\R^\R\colon \left|f^{-1}(y)\right|<\co\ 
           \text{ for every } y\in\R\right\},
\]
and, when $\co$ is a regular cardinal,
\[
{\mathcal{R}}_{in}=\left\{f\in\R^\R\colon \left|f[\R]\right|=\co\right\}.
\]
%In fact, the class 
%${\mathcal{R}}_{out}$ has the following nice characterization:
%\[{\mathcal{R}}_{out}=\left\{f\in\R^\R\colon \left|f^{-1}(y)\right|<\co\ 
%           \text{ for every } y\in\R\right\}.\]\label{pageRout}
In~\cite{CNat} the authors proved that

\newpage
%\medskip
\begin{theorem}[Ciesielski, Natkaniec~\cite{CNat}] %4.22
\label{th:CNatComp}
\
\begin{itemize}
\item[(1)] $\cin(\op{SZ})=2$.
\item[(2)] $\cout(\op{SZ})=\add(\op{SZ})$ 
           if $\co=\lambda^+$ for some cardinal $\lambda$.
\item[(3)] $\co<\cout(\op{SZ})\leq 2^\co$, if $\co$ is regular.
\item[(4)] $\cf(\co)\leq\cout(\op{SZ})\leq 2^{\cf(\co)}$, if $\co$ is singular.
\end{itemize} 
\end{theorem}

\medskip
Also, in a recent short survey paper~\cite{Nat:ShortSur} Natkaniec evaluated the
values of operators $\add$, $\mul$,
$\cin$ and $\cout$ for the class $\op{HAC}$ of {\em almost continuous functions
in sense of Husain}, i.e., such $f\colon\R\to\R$ that
$f^{-1}(U)\subset\op{int}(\cl(f^{-1}(U)))$ for every non-empty open set
$U\subset\R$. 

\medskip
\begin{theorem}[Natkaniec~\cite{Nat:ShortSur}]%4.23
\label{th:NatHus}
\
\begin{itemize}
\item[(1)] $\add(\op{HAC})=2^\co$.
\item[(2)] $\mul(\op{HAC})=\cov(\M)$ , where $\cov(\M)$ is the smallest
cardinality
           of a family of meager sets that covers $\R$.
\item[(3)] $\cin(\op{HAC})=\left(2^\co\right)^+$ and
           $\cout(\op{HAC})=\co$.
\end{itemize} 
\end{theorem}

\medskip
Some other cardinal operators connected with composition and
concerning some kind of coding were also studied by 
Ciesielski and Rec{\l }aw~\cite{CR},
Ciesielski and Natkaniec~\cite{CNat}, and
Natkaniec~\cite{Nat:ShortSur}.

Another variant of function $\add$ is connected  to the families of bounded
functions. To define it properly the following notation is necessary.  For a
Let 
$\op{UB}$  stand for all uniformly bounded families
${\mathcal{H}}\subset\R^\R$,  and let ${\mathcal{B}}$ be the class of all bounded
functions
$f\colon\R\to{\R}$. Then we define
$$
\begin{array}{rl} 
\add_b({\calF})\!\!\! & =\min\left\{\left|{\mathcal{H}}\right|\colon
{\mathcal{H}}\in\op{UB}\ \&\ 
\neg\left(\exists g\in{\mathcal{B}}\right) 
\left(\forall h\in{\mathcal{H}}\right)\left(g+h\in{\calF}\right)\right\}.
\end{array}
$$  
In~1994 Maliszewski~\cite{Ma} proved that
\[
\add_b({\mathcal{D}})=\cf(\co)
\] 
so that $\add_b({\mathcal{D}})<\add({\mathcal{D}})$.  
Moreover, he proved that if ${\cal F}\in\op{UB}$, all functions in
$\cal F$ are measurable (have Baire property), and the size of ${\cal
F}$ is less than the additivity of measure (category) then there
exists a ``universal summand'' bounded function for ${\cal F}$ with
the same property. Similar results were also proved for countable
families of Borel measurable functions of $\alpha$ class when
$\alpha>1$ and for finite families of Baire one functions.

The values of $\add_b$ for the other classes of functions from Chart~1 has been
investigated by Ciesielski and Maliszewski~\cite{CMa}. In particular, they proved

\newpage
%\medskip
\begin{theorem}[Ciesielski, Maliszewski~\cite{CMa}]%4.24
\label{th:CMal}
\
\begin{itemize}
 \item[(1)] $\add_b(\ext)=\add_b(\op{PR})=2$.
 \item[(2)] $\add_b(\ac) =\add_b(\conn) =\add_b(\D)=\cf(\co)$.
 \item[(3)] $\add_b(\pc)=\co$.
\end{itemize} 
\end{theorem}

\medskip \noindent
 Notice also that Theorem~\ref{th:CMal} implies immediately
the following  corollary. 

\medskip
\begin{corollary}[Ciesielski, Maliszewski~\cite{CMa}]%4.25
\label{cor:CMal}
\
\begin{itemize}
\item[(1)]  Every bounded function $f\colon\reals\to\reals$ is the sum of
 two bounded almost continuous functions.
\item[(2)] There exists a bounded function $f\colon\reals\to\reals$
 which is not the sum of two bound\-ed functions with perfect road.
\end{itemize} 
\end{corollary}

\medskip \noindent
 In particular, Corollary~\ref{cor:CMal}(1) generalizes  a
result of Darji and Humke~\cite{DH}  that every bounded function can be
expressed a sum of three  bounded almost continuous functions.  On the other
hand Corollary~\ref{cor:CMal}(2) shows that the following result of  Natkaniec
is sharp.

\medskip
\begin{theorem}[Natkaniec~\cite{Na2}] %4.26
\label{th:NatExt}
Every bounded function can be
expressed as a sum of three bounded extendable functions.
\end{theorem}

\medskip
It might be also interesting to examine a bounded version of $\mul$, defined as
$$
\begin{array}{rl} 
\mul_b({\calF})\!\!\! &  =\min\left\{\left|{\mathcal{H}}\right|\colon
{\mathcal{H}}\in
\op{UB}\ \&\ 
\neg\left(\exists g\in{\mathcal{B}},\, g\neq 0\right) 
\left(\forall h\in{\mathcal{H}}\right)\left(g\cdot h\in{\calF}\right)\right\}.
\end{array}
$$  
However this function has not been studied so far. 

One might also consider the study of the operator $\add$ (and $\mul$) for the
functions from $\real^n$ into $\real$ with $n>1$.  This has indeed been done by
Ciesielski and Wojciechowski in~\cite{CW}. The study concerned only the classes 
$\ext(\R^n)$, $\ac(\R^n)$, $\conn(\R^n)$, $\D(\R^n)$, and $\pc(\R^n)$ since
other classes from Chart~2 do not have natural generalizations into functions of
more than one variable.  First, one should recall that for $n>1$ Chart~1 is not
valid any more. The new inclusions (for $n>1$) are as follows:
$$
\ext(\R^n)\subset\pc(\R^n)=\conn(\R^n)\subset\D(\R^n)\cap\ac(\R^n),
$$
$$
\D(\R^n)\not\subset\ac(\R^n),\ \ \ 
\ac(\R^n)\not\subset\D(\R^n),\ \ \  \D(\R^n)\cap\ac(\R^n)\not\subset\conn(\R^n).
$$ (The inclusion ``$\pc(\R^n)\subset\conn(\R^n)$'' was proved by    
Hamilton~\cite{Hamilton} and by Stallings~\cite{Stal},  and the inclusion
``$\conn(\R^n)\subset\pc(\R^n)$'' by Hagan~\cite{Hagan}. The proof of the
inclusion  ``$\conn(\R^n)\subset\ac(\R^n)$'' 
is presented in~\cite{Stal}. The
examples showing that 
$\D(\R^n)\not\subset\ac(\R^n)$ and $\ac(\R^n)\not\subset\D(\R^n)$
can be found
in \cite[Examples 1.1.9 and 1.1.10]{N1hab} or 
\cite[Examples 1.7 and 1.6]{N1}, 
while a simple Baire class 1 function in
$\D(\R^n)\cap\ac(\R^n)\setminus\conn(\R^n)$
was described in 
\cite[Example~1]{RGR}.) We do not know whether the inclusion 
$\ext(\R^n)\subset\pc(\R^n)$ is proper. 

The problem with studying the value of the operator $\add$ for all  these
classes (except for $\ac(\R^n)$) is that  there exists a function
$f\colon\real^n\to\R$ which is not a sum  of $n$ Darboux functions, implying that
\[
\add(\ext(\R^n))=\add(\pc(\R^n))=\add(\conn(\R^n))=\add(\D(\R^n))=2.
\] 
However, every function $f\colon\R^n\to\R$ is a sum of $n+1$ extendable
functions. To express these results nicely, define for 
$\F\subset\R^{\R^n}$  the {\em repeatability} ${\mathcal{R}}({\cal F})$ of ${\cal
F}$  as the smallest integer $k$ such that any function 
$f\colon \R^n\rightarrow\R$ can be expressed as a sum of $k$ functions from
${\cal F}$. (We put ${\mathcal{R}}({\cal F})=\infty$ if such a number does not
exist.) In this language the results of  Ciesielski and Wojciechowski can be
stated as follows.

\medskip
\begin{theorem}[Ciesielski, Wojciechowski~\cite{CW}]%4.27
\label{th:CWoj}
\[
{\mathcal{R}}(\ext(\R^n))={\mathcal{R}}(\conn(\R^n))={\mathcal{R}}(\pc(\R^n))=n+1.
\] 
\end{theorem} 

\medskip \noindent
Clearly Theorem~\ref{th:CWoj} implies that ${\mathcal{R}}(\D(\R^n))\leq n+1$.
The problem (stated in~\cite{CW}) whether this equation can be replaced by the
equality has been recently solved by F.~Jordan.

\medskip
\begin{theorem}[F.~Jordan~\cite{FjordanPr}] %4.28
\label{th:JorDarb}
For every $n$ there exists
a Baire~1 class function $f\colon\R^n\to\R$ which is not a sum of $n$ Darboux
functions. In particular,
\[
{\mathcal{R}}(\D(\R^n))=n+1.
\] 
\end{theorem}

\medskip
The value of ${\mathcal{R}}(\ac(\R^n))$ is clearly equal to $2$, since
Natkaniec~\cite{N1} proved that $\add(\ac(\R^n))>\co$. This fact has been
recently improved by F.~Jordan, who proved 

\medskip
\begin{theorem} [F.~Jordan~\cite{FjordanPr}]%4.29
\label{th:JorACRn}
For every $n\geq 1$ 
\[
\add(\ac(\R^n))=\add(\ac)={\frak e}_\continuum.
\] 
\end{theorem}

\medskip \noindent
In~\cite{FjordanPr} 
F.~Jordan considers also the following version of additivity function
for the classes $\F$ of functions from $\R^n$ into $\R$
\begin{equation*}
\genadd_{n,k}(\F)=
\min\left(\left\{|G|\colon G\subseteq \real^{\R^n} \&\ \Psi_{n,k}(G) 
\text{ holds}\right\}
\cup\left\{\left(2^\continuum\right)^+\right\}\right)
\end{equation*}
where $k\F=\{f_0+\cdots+f_{k-1}\colon f_i\in\F\}$
and $\Psi_{n,k}(G)$ denotes the statement
\begin{equation*}
(\forall f\in (n-k)\F) 
(\exists g\in G)(g-f\notin (k+1)\F).
\end{equation*}
\newpage 
This function makes a good generalization of both
functions $\add$ and ${\mathcal{R}}$ since
\[
{\mathcal{R}}(\F) = n+1\ \text{ if and only if } \ 
1<\genadd_{n,k}(\F)<\left(2^\continuum\right)^+ \text{ for some } k<n,
\]
and
\[
\genadd_{1,0}(\F)+1=\add(\F)\ \ \text{ for any $\F\subseteq\R^\R$ 
such that $\F=\{-f\colon f\in\F\}$}.
\]
Thus, the following theorem generalizes Theorems \ref{th:CMill1}(a),
\ref{th:CRec}(1), \ref{th:CWoj}, and~\ref{th:JorDarb}.

\medskip
\begin{theorem} [F.~Jordan~\cite{FjordanPr}] %4.30
\label{th:JorNewest}
For every $n\geq 1$
\begin{itemize}
\item[(1)] $\genadd_{n,n-1}(\ext(\real^n))=\genadd_{n,n-1}(\conn(\real^n))=
\genadd_{n,n-1}(\pc(\real^n))=\co^+$;
\item[(2)] $\genadd_{n,n-1}(\D(\real^n))={\frak e}_\continuum$;
\item[(3)] $\genadd_{n,j}(\D(\real^n))=\add((j+1)\D(\R^n))=\continuum^+$
for any $j<n-1$ such that $2j\geq n-1$. 
\end{itemize}
\end{theorem}

\medskip
Notice also, that in the language of ${\mathcal{R}}$ operator the results from 
Theorem~\ref{th:NatExt} and Corollary~\ref{cor:CMal}(2) can be expressed by the
equation
\[
{\mathcal{R}}_b(\ext)={\mathcal{R}}_b(\SCIVP)=
{\mathcal{R}}_b(\CIVP)={\mathcal{R}}_b(\op{PR})=3,
\] 
where ${\mathcal{R}}_b$ is the natural generalization  of ${\mathcal{R}}$ for the
class of bounded functions. 




%\input{survey3}


\medskip
\stepcounter{section}
\section*{\bf 5. Some elements of topology}

\medskip
Let $X$ and $Y$ be arbitrary sets. For arbitrary families $\A\subset\P(X)$ and
$\B\subset\P(Y)$, where $\P(Z)$ stands for the collection of all subsets of a
set $Z$, define
\[
\C^{-1}_{\A,\B}=\{f\in Y^X\colon f^{-1}(B)\in\A\ \text{ for every }\ B\in\B\}
\] 
and 
\[
\C_{\A,\B}=\{f\in Y^X\colon f[A]\in\B\ \text{ for every }\ A\in\A\}.
\] 
If families $\A$ and $\B$ are the topologies on $X$ and $Y$, respectively,
then $\C^{-1}_{\A,\B}$ is a well known object: the class of all continuous
functions from $\la X,\A\ra$ to $\la Y,\B\ra$.  Similarly a class of measurable
functions  with respect to an algebra $\A$ of subsets of $X$ is equal to
$\C^{-1}_{\A,\B}$, where $\B$ is an appropriate topology 
\linebreak on $Y$. 

In both these approaches one  starts with families of sets
$\A$ and $\B$ and obtain, in return,  a family of functions. But what if a class
of functions $\F\subset Y^X$ is given to begin with? When can we find families
$\A\subset\P(X)$ and $\B\subset\P(Y)$ such that $\F=\C^{-1}_{\A,\B}$ or
$\F=\C_{\A,\B}$? And how nice can these families be, if they exist?

These questions have been studied recently by several authors. To talk about
their results, let us fix the following terminologies. We say that a family
$\F\subset Y^X$ can be
\begin{itemize}
\item {\em characterized by images of sets}\/ if $\F=\C_{\A,\B}$
      for some families $\A\subset\P(X)$ and $\B\subset\P(Y)$;
\item {\em characterized by preimages of sets}\/ if $\F=\C^{-1}_{\A,\B}$
      for some families $\A\subset\P(X)$ and $\B\subset\P(Y)$;
\item {\em topologized}\/ if $\F=\C^{-1}_{\A,\B}$
      for some topologies $\A$ on $X$ and $\B$ on $Y$;
\item {\em characterized by associated sets}\/ if $Y=\real$ and
      $\F=\C^{-1}_{\A,\B}$ for some family $\A\subset\P(X)$ 
      and $\B=\{(a,\infty)\colon a\in\real\}\cup\{(-\infty,b)\colon b\in\real\}$.
\end{itemize}

\bigskip
From all these notions only the problem of characterizing  by associated sets
has been extensively studied.  Clearly, all classes of continuous function 
$\C(X)$ from a topological space $X$ into $\R$  (considered with the natural
topology) can be characterized by associated sets. So can be the family of
$\B$-measurable functions from $X$ into $\R$, for any $\sigma$-algebra $\B$ of
subsets of $X$. However, there are also many examples of classes of functions
that do not admit such a characterization. In fact, the real interest in the
characterizations of functions by associated sets  has been initiated by
the~1950 paper of Zahorski~\cite{Za}, in which he tried to characterize
derivatives (from $\real$ to $\R$) in that way.%
\footnote{See also 1969 paper of Mr\'{o}wka~\cite{Mrowka} 
on characterizing functions by associated sets.}
Today we know that derivatives
cannot be characterized by associated sets: any class $\F$ that can be
characterized that way has the property that $h\circ f\in\F$ for every $f\in\F$
and  every homeomorphism
$h\colon\real\to\real$; however derivatives do not have this property. (See
Bruckner's book~\cite{Br} on this subject.  Compare also~\cite{Br2}.) This
negative result  has been followed by several others, in which the authors prove
that the following classes of functions (from $\R$ to $\R$) cannot be
characterized by associated sets:
$\D$ (Bruckner~\cite[1967]{AB}), 
$\conn$ (B.~Cristian, I.~Tevy~\cite[1980]{CT}), 
$\ac$ (Kellum~\cite[1982]{KK}), 
$\ext$ (Rosen~\cite[1996]{HR3}) and the remaining classes from Chart~2
(Ciesielski, Natkaniec~\cite[1997]{CNatImages}).

The question about topologizing different classes  of real functions has been
first systematically  studied in early 1990's by Ciesielski in~\cite{C}.%
\footnote{According to~\cite{PT} already in  a~1988 manuscript~\cite{Tar}
Tartaglia proved that the class of all derivatives cannot be topologized.} He
starts with the following theorem listing basic properties of classes that can
be topologized. In the theorem
$\complex$ stands for the set of complex numbers, 
$\Lin$ for the class of linear functions $f(x)=ax+b$,
$\ordinarytop$ for the natural topology on $\real$,  and $\id_X$ for the
identity function from $X$ to $X$. 

\newpage
%\medskip
\begin{theorem}[Ciesielski~\cite{C}]%5.1
\label{th:properties}
Let $\tau_1$ and $\tau_2$ be
the  topologies on sets $X$ and $Y$, respectively, and let
$\F=\C^{-1}_{\tau_1,\tau_2}\neq Y^X$. If
$\tau$ is the weak topology on $X$ generated by $\F$, 
that means,  generated by the family 
\linebreak $\{f^{-1}(U)\colon U\in\tau_2, f\in\F\}$, then
\begin{itemize}
\item[(i)]   $\Const\subset\F$, $\tau\subset\tau_1$, 
             $\tau_1\neq{\cal P}(X)$,
             $\tau_2\neq\{\emptyset,Y\}$ and $\F=\C^{-1}_{\tau,\tau_2}$;
\item[(ii)]  if $X=Y$ and\/ $\id_X\in\F$ 
             then $\tau_2\subset\tau\subset\tau_1$;
\item[(iii)] if $Y=\reals$ and $\ordinarytop\subset\tau_2$ then $\F$ is closed 
             under the maximum and minimum operations;
\item[(iv)]  if $\G\subset Y^Y$ is such that\/
             $\id_Y\in\G$ and $g\circ f\in\F$ 
             for all $f\in\F$ and $g\in\G$ then 
             $\F=\C^{-1}_{\tau,\tau^\prime}$,
             where $\tau^\prime$ is a topology generated by 
             $\{g^{-1}(U)\colon U\in\tau_2, g\in\G\}$; 

             in particular, if $\G=\Lin$ than we may assume that
             $\tau_2$ is a homothetically closed $T_1$ topology;
\item[(v)]   if $X=Y$, $\id_X\in\F$ and $\F$ is closed under the
             composition,
             then $\F=\C^{-1}_{\tau,\tau}$;
\item[(vi)]  if $X=Y\in\{\reals,\complexnumb\}$ and $\Lin\subset\F$
             then $\tau_1$ is a $T_1$ topology;
\item[(vii)] if $X=Y\in\{\reals,\complexnumb\}$, $\Lin\subset\F$
             and $\tau_2$ contains two nonempty disjoint sets, 
             then $\tau_1$ is Hausdorff;
\item[(viii)] if $X=Y=\reals$ and every $f\in\F$ 
             is Darboux then $\tau_1$ is connected;
\item[(ix)]  if $X=Y=\reals$, $\Lin\subset\F$ and $\tau_2$ contains a 
             non-empty set
             which has either upper or lower bound, then 
             $\ordinarytop\subset\tau_1$.
\end{itemize} 
\end{theorem}

\smallskip \noindent
 Of all these properties only (iii) needs a little longer
(but still easy) argument. Note also, that (i) shows, that in order to
topologize some family, only the search for the range topology is essential.
Condition (v) shows that the question when topologies
$\tau_1$ and $\tau_2$ can be chosen equal is answered by the following
corollary. 

\smallskip
\begin{corollary} [Ciesielski~\cite{C}] %5.2
\label{cor:sameTop}
Let $\F\subset X^X$. If $\F$ can
be topologized then $\F=\C_{\tau,\tau}$ for some topology $\tau$ on $X$ if and
only if\/ $\id_X\in\F$ and 
$\F$ if closed under the composition operation.
\end{corollary}

\smallskip
Next, from Theorem~\ref{th:properties} (conditions (iii), (vi) and (ix))
Ciesielski  concludes the following fact

\smallskip
\begin{theorem}[Ciesielski~\cite{C}] %5.3
\label{th:Cinfinity} 
Let $\F$ be a family of real
functions closed under composition and such that 
$\C^\infty\subset\F$. If $\F$ can be topologized then $\F$ is closed under  the
maximum and minimum operations. 
\end{theorem}

\smallskip \noindent
 which easily leads to the following corollary:

\smallskip
\begin{corollary} [Ciesielski~\cite{C}]%5.4
\label{cor:Cinf}
 The classes: $\C^\infty$  of
infinitely many times differentiable  functions, ${\cal D}^n$ of $n$-times
differentiable  functions, and $\C^n$ of functions with continuous $n$-th 
derivative cannot be topologized. The same is true, when in the above we replace
differentiability with symmetric differentiability, approximate
differentiability, symmetric approximate differentiability, 
$\I$-approximate differentiability or  symmetric $\I$-approximate
differentiability.
\end{corollary}

\medskip \noindent
(The definitions of all classes of functions from this, and the next corollary 
can be found in~\cite{Br} and in~\cite{CLO:book}.)

With a little more effort he also concludes

\medskip
\begin{corollary} [Ciesielski~\cite{C}]%5.5
\label{cor:der}
The following classes cannot be
topologized:  the class $\Delta$ of all derivatives,% 
\footnote{See also~\cite{Tar}.} the Zahorski's classes ${\cal M}_i$ for
$i=0,1,2,3,4$,  the class of all  symmetrically  (symmetrically approximately or
symmetrically $\I$-approximately)  continuous functions, the class of all
Darboux functions, the class of all measurable functions and the class of all
functions having the Baire property.
\end{corollary}

\medskip
From the positive side, paper~\cite{C} contains the following deeper result.

\medskip
\begin{theorem}[Ciesielski~\cite{C}]%5.6
\label{th:TopMain}
 Let $|X|=|Y|=\continuum$,
${\mathcal{R}}\subset Y^X$ be of cardinality
$\leq\co$ and let $\I$ be a proper $\sigma$-ideal on $X$ containing all
singletons.
\begin{itemize} 
\item[(A)] If GCH holds then there is a Hausdorff, connected and locally
connected topology $\tau_2$ on $Y$ with the property that for every family
$\F\subset\Const\cup{\mathcal{R}}$  such that $\Const\subset\F$ and
\[\{x\in X\colon f(x)=g(x)\}\in\I\ \text{ for every distinct }\ f,g\in\F 
\] we have
\[
\F=\C^{-1}_{\tau,\tau_2}, 
\] where $\tau$ is generated by the family
$\{f^{-1}(U)\colon U\in\tau_2\ \&\ f\in\F\}$.  Topology $\tau$ is connected and
locally connected. It is also Hausdorff,  provided $\F$ separates points.
\item[(B)]  Moreover, it is consistent with the set theory ZFC+GCH  that the
topologies $\tau$ and $\tau_2$ are completely regular and Baire. 
\end{itemize}  
\end{theorem}

\medskip
Applying Theorem~\ref{th:TopMain} to the 
$\sigma$-ideal $\M$ of the first category  subsets of $\reals^n$, and using  the
fact that for any different harmonic functions
$f,g\colon \reals^n\to\reals^m$ we have 
${\rm int}_{\tau_{\cal O}}(\{x\in X\colon f(x)=g(x)\})=\emptyset$ we can
conclude that the class of all harmonic functions
$f\colon \reals^n\to\reals^m$ can be topologized.

Another $\sigma$-ideal that can be used with Theorem~\ref{th:TopMain}  is the
ideal $\I_\omega$ of at most countable sets.  Since for any two different 
analytic functions $f,g\in\cal A$ we have 
$\{x\colon f(x)=g(x)\}\in\I_\omega$, we can also conclude the following
corollary. 

\medskip
\begin{corollary}[Ciesielski~\cite{C}]%5.7
\label{cor:mainB}
 If GCH holds  then there is a
Hausdorff, connected and locally connected topology 
$\tau_{\cal A}$ (on\/ $\reals$ or\/ $\complexnumb$) such that for any family 
$\F\supset\Const$ of analytic functions we have
\[
\F=\C^{-1}_{\tau_{\cal F},\tau_{\cal A}},
\] where $\tau_{\cal F}$ is generated by the family $\{f^{-1}(U)\colon
U\in\tau_{\cal A}\ \&\ f\in\F\}$. Moreover, $\tau_{\cal F}$ is connected and
locally connected, and it is Hausdorff provided $\F$ separates points. 

It is also consistent with ZFC+GCH that all these topologies are  completely
regular and Baire. 
\end{corollary}

\medskip
Notice also, that if the family $\F$  in Corollary~\ref{cor:mainB} is closed
under the composition and
$\id\in\F$, then, by Theorem~\ref{th:properties}(v),
$\F=\C^{-1}_{\tau_{\cal F},\tau_{\cal F}}$. We can write this in the form of next
corollary, where $\A$ stands for the family of all analytic functions and ${\cal
P}$ for the family of all polynomials. 

\medskip
\begin{corollary}[Ciesielski~\cite{C}] %5.8
\label{cor:mainC}
If GCH holds  and $\F$ is a  family
of real  functions which is closed under the composition and such that 
$\{\id\}\cup\Const\subset\F\subset\A$, then there exists a Hausdorff,  connected
and locally connected topology $\T_{\cal F}$  (on $\reals$ or $\complexnumb$)
such that $\F=\C^{-1}_{\T_{\cal F},\T_{\cal F}}$. 
In particular, there exist a  ``linear topology'' $\T_{\cal L}$, a
``polynomial topology'' $\T_{\cal P}$ and an ``analytic topology'' 
$\T_{\cal A}$ which are Hausdorff, connected and locally connected such that 
$\T_{\cal L}\subset\T_{\cal P}\subset\T_{\cal A}$ and for which 
$\Lin=\C^{-1}_{\T_{\cal L},\T_{\cal L}}$, 
${\cal P}=\C^{-1}_{\T_{\cal P},\T_{\cal P}}$, and 
$\A=\C^{-1}_{\T_{\cal A},\T_{\cal A}}$.

Moreover, it is  consistent with ZFC+GCH that all these topologies are 
completely regular and Baire. 
\end{corollary}

\medskip
The following questions in these subject are open.

\medskip
\begin{problem}[Ciesielski~\cite{C}]%10
\label{pr:ZFC}
\ 
{\em
\begin{itemize} 
\item[(1)]  Can we prove Theorem~\ref{th:TopMain} or any of the 
Corollaries~\ref{cor:mainB}, or~\ref{cor:mainC} without any additional
set-theoretical assumptions?
\item[(2)] Can topologies from  Theorem~\ref{th:TopMain} or any of the 
Corollaries~\ref{cor:mainB}, or~\ref{cor:mainC} be normal? Lindel\"of?
hereditarily Lindel\"of? compact? metrizable?
\end{itemize} }
\end{problem}

\medskip
The general problem of characterizing classes of functions by preimages of sets
(in a sense defined above)  has been studied only in two papers: \cite{KC:Der}
and~\cite{CNatImages}. In paper~\cite{KC:Der} Ciesielski  proves the following
theorem, which generalizes a similar result of Preiss and Tartaglia~\cite{PT}.

\newpage
%\medskip
\begin{theorem} [Ciesielski~\cite{KC:Der}]%5.9
\label{th:DerMain}
 Let $\F\subset\real^\real$ be
a family of cardinality less than or equal to $\continuum^+$ and let
$\G\subset\real^\real$ be such that
\begin{itemize}
\item[(1)] $\G$ contains all constant functions;
\item[(2)] $|\G|\leq\continuum$; and,
\item[(3)] $|f[\real]|=\continuum$ for any non-constant $f\in\real^\real$ which
is a difference of two functions from $\G$.
\end{itemize} Then there exists a family
$\A\subset\P(\real)$ of cardinality less than or equal to $|\F|$ such that
\[
\G\cap\F=\C_{\D,\A}\cap\F
\] where $\D=\{f^{-1}(A)\colon A\in\A\ \&\ f\in\G\}$. 
\end{theorem}

\medskip \noindent
Clearly the family
$\G=\Delta$ of all derivatives satisfies the above conditions (1)--(3). 
Thus, using the theorem with $\G=\Delta$
and $\F$ equal to the family $\operatorname{{\cal B}or}$
of Borel functions we obtain the following corollary. 

\medskip
\begin{corollary} %5.10
\label{corDouble}
There exists a family $\A\subset\P(\real)$ such that 
$|\A|\leq\co$ and
\[
\Delta=\operatorname{{\cal B}or} \cap \, \C (\D ,\A ),
\]
where $\D=\{f^{-1}(A)\colon f\in\Delta\ \&\ A\in\A\}$.
\end{corollary}

\medskip \noindent
However, the following stronger characterization of
$\Delta$ is also described in~\cite{KC:Der},
where $\D\B_1$ stands for the class of Darboux Baire one functions.

\medskip
\begin{theorem}%5.11
\label{thZFCdel}
There exists a Bernstein set $B\subset\real$ such that 
\[
\Delta=\D\B_1\cap\C(\D_0,\{B+c\colon c\in\real\})=\C(\D,\A),
\]
where $\A=\bigcup_{c\in\real}\{(-\infty,c),(c,\infty),B+c\}$,
$\D=\{f^{-1}(A)\colon f\in\Delta\ \&\ A\in\A\}$, and
$\D_0=\{f^{-1}(B+c)\colon f\in\Delta\ \&\ c\in\real\}$.
\end{theorem}

\medskip
Note that by Corollary~\ref{cor:der} the families $\D$ and $\A$ in 
Theorem~\ref{thZFCdel} cannot be topologies.  Also, they cannot be algebras:

\medskip
\begin{theorem}[Ciesielski~\cite{KC:Der}]%5.12
\label{th:KCDerAlg}
 If $\Delta=\C^{-1}_{\D,\A}$
for some families 
$\D$ and $\A$ of subsets of\/ $\real$ then neither $\A$ nor $\D$  contain
simultaneously a non-empty proper subset 
$S$ of\/ $\real$ and its complement\/ $\real\setminus S$.

In particular, neither $\A$ nor $\D$ is an algebra.
\end{theorem}

\medskip
The following problem remains open.

\medskip
\begin{problem}[Ciesielski~\cite{KC:Der}]%11
\label{prDer}
{\em Can the family $\A$ in Corollary~\ref{corDouble} or
Theorem~\ref{thZFCdel} consist of any kind of regular sets like Lebesgue
measurable, Borel, or sets with Baire property?}
\end{problem}

\medskip
An interesting discussion concerning characterizations
of the derivatives can be 
also found in a recent article of Freiling~\cite{Freil}.

\medskip
The problem of characterizing by preimages of sets  families from Chart~2 has
been recently addressed by Ciesielski and Natkaniec.

\medskip
\begin{theorem}[Ciesielski, Natkaniec~\cite{CNatImages}] %5.13
\label{th:NatPreim}
\
\begin{itemize} 
\item[(1)] The classes: $\op{SZ}$,
$\ext$, $\ac$, $\conn$, $\D$, $\SCIVP$,
$\CIVP$, and $\WCIVP$ cannot be characterized by preimages of sets. 
\item[(2)] The classes: $\op{PR}$ and $\pc$ can be characterized by preimages of
sets as $\C^{-1}_{\A,\B}$ with $\B$ being the natural topology on $\R$. However,
they can neither be topologized nor be characterized by associated sets. 
\end{itemize}  
\end{theorem}

\medskip
The problem of characterizing a family of functions by images of sets was first
studied by  Velleman for the class $\C$ of continuous functions from $\R$ 
\linebreak to $\R$.

\medskip
\begin{theorem}[Velleman~\cite{Vel}]%5.14
\label{th:Vel}
\
\begin{itemize} 
\item[(1)] $\C=\C_{\A,\A}\cap\C_{\B,\B}$, where
$\A$ is the family of all connected subsets of\/ $\R$ and 
$\B$ the family of all compact subsets of\/ $\R$.
\item[(2)] $\C$ cannot be characterized by images of sets.
\end{itemize} 
\end{theorem}

\medskip \noindent
 Note that a family $\C_{\A,\A}$ from Theorem~\ref{th:Vel}
is just the family $\D$ of Darboux functions. 

Theorem~\ref{th:Vel}(1) has been recently generalized 
by Arenas and Puertas~\cite{ArPu}. 
Theorem~\ref{th:Vel}(2) has been essentially generalized by Ciesielski,
Dikranjan and Watson in~\cite{CDW}. In this paper the authors  list a basic
properties of classes that can be characterized  by images of sets, which is
similar in flavor to  Theorem~\ref{th:properties}. Then, they prove the
following generalization of Theorem~\ref{th:Vel}.

\medskip
\begin{theorem}[Ciesielski, Dikranjan, Watson~\cite{CDW}]%5.15
\label{th:CDW1}
 For a Ty\-cho\-noff topological space $X$ the class 
$\F=\C(X,\real)$ of all continuous functions from
$X$ to $\R$ can be characterized by images of sets if and only if $X$ is a
discrete space. 
\end{theorem}

\medskip
They also remarked that there is a compact subset $K\subset\R^2$, a Cook
continuum, for which $\C(K,K)=\Const\cup\{\id_K\}$, and so, it can be 
characterized by images of sets. 

For the classes of functions from $\R$ to $\R$, their generalization of
Theorem~\ref{th:Vel} appears as follows.

\medskip
\begin{theorem}[Ciesielski, Dikranjan, Watson~\cite{CDW}]%5.16
\label{th:mainR}
If
$\A,\B\su\P(\real)$ are such that $\C\su\C_{\A,\B}$ then there is a
non-measurable function $f\in \C_{\A,\B}$.
\end{theorem}

\medskip
This, in particular, implies the following corollary.

\medskip
\begin{corollary}[Ciesielski, Dikranjan, Watson~\cite{CDW}]%5.17
\label{cor:mainR}
Neither of
the following classes of functions from $\real$ to
$\real$ can be represented as $\C_{\A,\B}$ for any $\A,\B\su\P(\real)$:
\begin{itemize}
\item the class of upper or lower semicontinuous functions;
\item the class $\Delta$ of derivatives;
\item the class of approximately continuous functions;
\item the class of Baire class~1 functions;
\item the class of Borel functions;
\item the class of measurable functions. 
\end{itemize} 
\end{corollary}

\medskip \noindent
 They also noticed that the class $\D$ of Darboux functions  can
be characterized by images of sets. (It is defined that way.)

It has been also recently noticed by 
Ciesielski and Natkaniec~\cite{CNatImages}
that in Theorem~\ref{th:mainR}
the clause ``non-measurable'' cannot be replaced by ``without the Baire
property.'' More precisely, they proved 
 
\medskip
\begin{theorem} [Ciesielski, Natkaniec~\cite{CNatImages}] %5.18
\label{th:KCdivp}
Let
$f\in\C_{\A,\A}$, where
\[
\A=\{D\cap I\colon D\ 
\mbox{ is dense in $\real$ and $I\subset\real$ is an interval }\}.
\] 
Then $f$ is continuous on a dense set.

In particular $f$ has the Baire property and 
\[
\C\subset\C_{\A,\A}\subset\B aire,
\] 
where 
$\B aire$ stands for the class of functions $g\colon\R\to\R$  with the Baire
property. 
\end{theorem}

\medskip
Finally, Ciesielski and Natkaniec~\cite{CNatImages}
proved that it is impossible to characterize
by images of sets the classes $\op{SZ}$, and ${\cal B}aire$ of 
functions (from $\R$ \linebreak to $\R$) with the Baire
property. They also proved the following theorem. 

\medskip
\begin{theorem} [Ciesielski, Natkaniec~\cite{CNatImages}]%5.19
\label{th:NatPreim2} 
\ \ The\ \  classes:\ \  $\CIVP$, \linebreak $\WCIVP$, and
$\D$ can be characterized by images of sets. However, the
remaining classes from Chart~2 cannot be represented that way.
\end{theorem}

\medskip
The following problem in this area remains open.

\newpage
%\medskip
\begin{problem} [Ciesielski, Dikranjan, Watson~\cite{CDW}]%12
\label{prReal}
 Can any of the \linebreak
follow\-ing classes of real functions be represented as $\C_{\A,\B}$?
\begin{itemize}
\item The class of all linear functions $f(x)=ax+b$. 
\item The class of all polynomials.
\item The class of all real analytic functions.
\item The class $C^\infty$ 
      of infinitely many times differentiable functions.
\item The class $D^n$ of $n$-times differentiable functions, 
      with \linebreak $1\leq n<\omega$. 
\end{itemize} 
\end{problem}

\medskip
Another interesting problem (loosely related to real functions, but having the
same flavor that the topologizing question has)  concerns the existence of a
topology on a given set
$X$, often the real line, satisfying the best possible separation axioms,  for
which a given ideal ($\sigma $-ideal) of  subsets of $X$ consists precisely of
sets that are nowhere dense  (or first category) in $X$.   Ciesielski and
Jasinski \cite[1995]{CJ}  obtained several positive results in this direction 
under some additional set-theoretic assumptions.   The problem was also
investigated  in the papers~\cite{Ro} by Rogowska and~\cite{BRo} by Balcerzak
and Rogowska.

There are also many interesting theorems concerning  different classes of
functions $f\colon\real^n\to\real$, where
$\R^n$ is equipped with some abstract topology refining  of the natural
topology. A survey of some recent results in this direction can be found in the
last issue of the Real Analysis  Exchange~\cite{GKSW}.  The topologies on $\real$ 
that were
most studied in this aspect in recent years are the $\I$-density 
topology (defined in 1982 by Wilczy{\'n}ski~\cite{Wil:Def})
and the deep
$\I$-density topologies 
(defined in 1986 independently by {\L}azarow~\cite{Laz}, and by 
Poreda and  Wagner-Bojakowska~\cite{PoWB}).
These are category analogues of the density
topology. The survey of the results in this direction can be found 
in a monograph of Ciesielski, Larson and Ostaszewski~\cite{CLO:book}. 
(In particular, see~\cite{CL1} or~\cite[Sec.~1.5]{CLO:book}
for some set theoretic results and open problems concerning these topologies.) 

\medskip
\stepcounter{section}
\section*{\bf 6. Elements of measure theory} 

\medskip
The Lebesgue measure, being a function from family of subsets
${\cal L}$ of $\R^n$ into $[0,\infty]$, is not of the form
$f\colon\real^n\to\R$, so it does not lie directly in a scope of this article.
However, it is certainly one of the main tools of  real analysis and many
results concerning its generalizations  have a deep set theoretical context.
Therefore,  a short section concerning 
the newest developments in this area has been
added to this  paper.

An accessible survey concerning different  extensions of Lebesgue
measure can be found in  the~1989 Mathematical Intelligencer
article~\cite{KC:HowGood}  of K.~Ciesielski. The best survey concerning
universal (i.e., defined on $\P(\real)$) countable additive extensions of
Lebesgue measure can be found in the~1993 survey article of
D.~H.~Fremlin~\cite{FrRVMC}. Thus, we will concentrate here only on the newest
developments, that concern isometrically invariant extensions of Lebesgue
measure. (See also M.~Laczkovich survey article~\cite{Lacz} on this subject.)

Recall here, that by the 1923 theorem of Banach there is a finitely additive
isometrically invariant measure $\mu\colon\P(\R^2)\to[0,\infty]$ extending
Lebesgue measure, while such a measure on $\R^3$  does not exist by a famous
Banach-Tarski Paradox (1924):
\begin{itemize} 
\item the ball $B\subset\real^3$ and  the cube $Q\subset\real^3$ (of arbitrary
volumes) are isometrically equivalent, i.e., there  is a finite partition
$\{B_k\}_{k=1}^n$ of $B$ and isometries
$\{i_k\}_{k=1}^n$ of $\real^3$ such that
$\{i_k[B_k]\}_{k=1}^n$ forms a partition of $Q$. 
\end{itemize}  There were two famous problems around this subject. The first
one, due to Marczewski, was whether the pieces
$\{B_k\}_{k=1}^n$ in the Banach-Tarski Paradox can have the Baire property.  The
answer to this question, surprisingly positive, was obtained  by Dougherty and
Foreman in~1994.

\medskip
\begin{theorem}[Dougherty, Foreman~\cite{DF}] %6.1
\label{thDF}
 For any $n\geq 3$ any two
bounded  non-meager sets $B,Q\subset\R^n$ with the Baire property  are
isometrically equivalent with pieces having the Baire property.
\end{theorem}

\medskip
The second famous question was the Tarski's circle-squaring problem: {\em is a
circle $C\subset\real^2$ of the unit area equivalent to a square
$S\subset\real^2$ of the unit area?} Note that if the areas of $C$ and $D$ were
different, then Banach's theorem of~1923 would immediately imply the
negative  answer. However, the answer to Tarski's circle-squaring problem is
positive, as proved by Laczkovich in~1990.

\medskip
\begin{theorem}[Laczkovich~\cite{LaSqCir}] %6.2
\label{thSqCir}
Any two sets $B,Q\subset\R^n$ 
having the same area and being bounded by Jordan curves are   isometrically
equivalent.
\end{theorem}

\medskip
The other class of isometrically invariant extensions of Lebesgue measures on
$\R^n$  concerns countably additive extensions.  In 1936 Sierpi\'nski asked,
whether such an extension can be maximal. The negative answer to this question 
was given in~1977 by A.~B.~Kharazishvili~\cite{Khar} (for $n=1$) and in~1985 by
Ciesielski and Pelc~\cite{KC:Sierpinski1} (for arbitrary $n$). (Compare also
\cite{C:a,C:b,K-Z,Z1}.)

\medskip
\begin{theorem} [Ciesielski, Pelc~\cite{KC:Sierpinski1}] %6.3
\label{th:CP}For every $n\geq 1$
and eve\-ry count\-ably additive  isomatrically invariant extension $\mu\colon{\cal
M}\to[0,+\infty]$ of the Lebesgue measure on $\R^n$ there exists  a proper
countably additive  isomatrically invariant extension $\mu'\colon{\cal
M}'\to[0,+\infty]$ of $\mu$.
\end{theorem}

\medskip \noindent
A weak side of this theorem was that the extension $\mu'$ of
$\mu$ was only by (new) measure zero sets so, in a way, trivial. This has been
recently improved by Zakrzewski, who showed

\medskip
\begin{theorem} [Zakrzewski~\cite{Zak}]%6.4
\label{th:Zak}
For every $n\geq 1$ and every
countably additive  isomatrically invariant extension $\mu\colon{\cal
M}\to[0,+\infty]$ of the Lebesgue measure on $\R^n$ there exists  a countably
additive  isomatrically invariant extension $\mu'\colon{\cal M}'\to[0,+\infty]$
of $\mu$ such that the canonical embedding $\op{e}\colon{{\cal M}/\mu}\to{{\cal
M}'/\mu'}$ of measure algebras defined by $\op{e}([A]_{\mu})=[A]_{\mu'}$ is not
surjective. 
\end{theorem}

\medskip \noindent
Zakrzewski's proof is based on a construction of Kharazishvili~\cite{KharNew}
from 1997, which was known earlier to imply 
Theorem~\ref{th:Zak} for $n=1$. 

Many interesting results concerning %isometrically 
invariant extensions of Lebesgue measure can be also found in~1983 
book of Kharazishvili~\cite{Kh:Book}.



%\input{survey4}

%\input{polish}
\bigskip %\bigskip
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\end{thebibliography}



\bigskip %\bigskip 
\noindent {\small \sc
{Krzysztof Ciesielski\\
Department of Mathematics\\
West Virginia University\\ Morgantown, WV 26506-6310 USA\\
KCies\AT wvnvms.wvnet.edu 
}}

\end{document}

Krzysztof Chris Ciesielski
Department of Mathematics, West Virginia University 
Morgantown, WV 26506-6310
phone: (304) 293-2011 ext. 2337
FAX: (304) 293-3982 
http://www.math.wvu.edu/homepages/kcies/

