\documentclass{article}
\usepackage{encyclopedia}
\usepackage{amssymb}
\usepackage{amsmath}



\title{Generalized continuities}
\author{Krzysztof Ciesielski}
\address{West Virginia University,
Morgantown, WV 26506-6310, USA\\
e-mail: {\tt K\_Cies@math.wvu.edu};
{\tt http://www.math.wvu.edu/\~{}kcies}
}
\date{May 14, 2001}

%Two macros

   \newcommand{\real}{{\mathbb R}}

% characteristic function
   \newcommand{\charf}[1]{\mbox{\raise.48ex\hbox{$\chi$}$_{#1}$}}




\begin{document}

\maketitle

There are many different properties of functions that can be considered
as generalized continuity notions. In this article we will describe
mainly those that arose from the study of real functions, concentrating
first on the classes
of functions known under the common name of
Darboux-like functions.
(See e.g. survey articles \cite{GN,GNup,CJas,KCsurv}.)

For topological spaces $X$ and $Y$, 
a function $f\colon X\to Y$ is \defterm{Darboux},
$f\in{\rm D}(X,Y)$,
provided the image $f[C]$ of $C$ under $f$ is a
\otherterm{connected} subset of $Y$
for every connected subset $C$ of $X$.
In particular, $f\colon\real\to\real$ is Darboux
provided $f$ maps intervals onto intervals, that is,
when it has the \otherterm{intermediate value property}.
The name comes after G.~Darboux who showed in 1875 that
every derivative (of a function from $\real$ to $\real$)
has the intermediate value property,
while there are derivatives discontinuous on a
\otherterm{dense set}.
(Some 19th century mathematicians thought that the intermediate value property
could be taken as the definition of continuity. Some calculus
teachers still think so.)
One of the easiest examples of a discontinuous Darboux function is
$f_0\colon\real\to\real$ given by
$f_0(x)=\sin(1/x)$ for $x\neq 0$ and $f_0(0)=0$.

A function $f\colon X\to Y$ is \defterm{connectivity},
$f\in{\rm Conn}(X,Y)$,
if the graph of the restriction $f\restriction Z$ of $f$ to $Z$
is connected in $X\times Y$ for every connected subset $Z$ of~$X$.
It is easy to see that $f\colon\real\to\real$ is connectivity
if and only if its graph is a connected subset of $\real^2$.
However, there are functions $F\colon\real^2\to\real$
with a connected graph
which are not connectivity functions. For example, this is the case if
$F(x,y)=\sin(1/x)$ for $x\neq 0$, and $F(0,y)=h(y)$, where
$h\colon\real\to[-1,1]$ is any function with a disconnected graph.

A function $f\colon X\to Y$ is 
\defterm{extendable}, $f\in{\rm Ext}(X,Y)$, provided
there exists a connectivity function $F\colon X\times[0,1]\to Y$
such that $f(x)=F(x,0)$ for every $x\in X$.
It is easy to see that
\begin{center}\vspace{-2pt}
${\rm C}(X,Y)\subset{\rm Ext}(X,Y)\subset{\rm Conn}(X,Y)\subset{\rm D}(X,Y)$
\end{center}\vspace{-2pt}
for arbitrary topological spaces, where ${\rm C}(X,Y)$ stands for
the class of all \otherterm{continuous functions} from $X$ into $Y$.

A function $f\colon X\to Y$ is
\defterm{almost continuous} (in the sense of Stallings),
$f\in{\rm AC}(X,Y)$,
provided
each open subset of $X\times Y$ containing the graph of
$f$ also contains the graph of a continuous function from $X$ to $Y$.
This property was defined as a generalization of functions
having the \otherterm{fixed point property}.
It is easy to see that if every function in
${\rm C}(X,X)$ has the fixed point property, then so does
every $f\in{\rm AC}(X,X)$.

A function $f\colon X\to Y$ is
\defterm{peripherally continuous}, $f\in{\rm PC}(X,Y)$,
if for every $x\in X$ and for all pairs of open sets
$U$ and $V$ containing
$x$ and $f(x)$, respectively, there exists an open subset $W$ of $U$
such that $x \in W$ and $f[{\rm bd}(W)]\subset V$,
where ${\rm bd}(W)$ is the \otherterm{boundary} of $W$.
For the functions $f\colon\real\to\real$
this means that $f$ has the \defterm{Young property},
that is, for every
$x\in\real$ there exist sequences $\{x_n\}_n$ and $\{y_n\}_n$ such
that $x_n\nearrow x$, $y_n\searrow x$, and both $f(x_n)$ and
$f(y_n)$ converge to $f(x)$.
In 1907 J.~Young showed that
for the \otherterm{Baire class~1} functions,
the Darboux property and the Young property are equivalent.

We will discuss the above mentioned classes only when $Y=\real$.
If $X=\real^n$ and $n>1$ the relations between these classes are
given by the following chart, where arrows
$\longrightarrow$ denote strict inclusions. (See \cite{CJas}.)


\vspace{-10pt}
\hskip3pc\begin{picture}(0,90)
\put(20,55){\makebox(0,0){${\rm C}$}}
\put(30,55){\vector(1,0){15}}
\put(90,55){\makebox(0,0){${\rm Ext}={\rm Conn}={\rm PC}$}}
\put(138,55){\vector(1,0){15}}
\put(180,55){\makebox(0,0){${\rm AC}\cap{\rm D}$}}
   \put(205,60){\vector(2,1){18}}
   \put(237,70){\makebox(0,0){${\rm AC}$}}
   \put(237,40){\makebox(0,0){${{\rm D}}$}}
   \put(205,52){\vector(2,-1){18}}
\end{picture}
\vspace{-3pc}\vspace{-6pt}
\begin{center} Chart~1: Darboux-like functions from $\real^n$, $n>1$,
into $\real$.
\end{center}
The inclusion  ${\rm Conn}\subset{\rm Ext}$
was proved by
K.~Ciesielski, T.~Natkaniec, and J.~Wojciechowski~\cite{CNW}.
The containment ${\rm Conn}\subset{\rm PC}$ was proved by
O.H.~Hamilton and J.~Stallings, and the inclusion
${\rm PC}\subset{\rm Conn}$ by M.R.~Hagan.
The relation ${\rm Conn}\subset{\rm AC}$ was proved by
J.~Stallings.
It is important  to notice that Chart~1 remains unchanged
if we consider only Baire class~1 functions~\cite{CJas}.

Classes presented in Chart~1 were also studied within the class
${\rm Add}(\real^n)$ of \defterm{additive functions}, that is, functions
$f\colon\real^n\to\real$ for which
$f(x+y)=f(x)+f(y)$ for all $x,y\in\real^n$.
In this case Chart~1 transforms to

\vspace{-10pt}
\hskip1.5pc\begin{picture}(0,90)
\put(125,55){\makebox(0,0){${\rm C}={\rm Ext}=
{\rm Conn}={\rm PC}={\rm AC}\cap{\rm D}$}}
   \put(205,60){\vector(2,1){18}}
   \put(237,70){\makebox(0,0){${\rm AC}$}}
   \put(237,40){\makebox(0,0){${{\rm D}}$}}
   \put(205,52){\vector(2,-1){18}}
\end{picture}
\vspace{-3pc}\vspace{-6pt}
\begin{center} Chart~2: Additive
Darboux-like functions from $\real^n$, $n>1$, into $\real$.
\end{center}
The inclusion ${\rm AC}\cap{\rm D}\subset{\rm C}$
was proved by K.~Ciesielski and J.~Jastrz{\c{e}}bski~\cite{CJas}.


The Darboux-like functions were most intensively studied
when $X=Y=\real$. In this setting, more classes 
are considered Darboux-like.
Thus, a function $f\colon\real\to\real$ has:
the \defterm{Cantor intermediate value property}
if for every $x,y\in\real$ and for each \otherterm{perfect set} $K$
between $f(x)$ and $f(y)$
there is a perfect set $C$
between $x$ and $y$ such that $f[C]\subset K$;
the \defterm{strong Cantor intermediate value property} if
for every $x,y\in\real$ and for each perfect set $K$
between $f(x)$ and $f(y)$
 there is a perfect set $C$ between $x$ and $y$ such that
 $f[C]\subset K$ and $f\restriction C$ is continuous;
the \defterm{weak Cantor intermediate value property}
if for every $x,y\in\real$
with $f(x)<f(y)$ there exists a perfect set $C$ between
$x$ and $y$ such that $f[C]\subset (f(x),f(y))$;
the \defterm{perfect road} if for every $x\in\real$
there exists a perfect set
$P\subset\real$ having $x$ as a bilateral (i.e., two sided)
limit point for which $f\restriction P$ is continuous at $x$.
The classes of these functions are denoted by ${\rm CIVP}$,
${\rm SCIVP}$, ${\rm WCIVP}$,
and ${\rm PR}$, respectively. 
The relations between them are as follows~\cite{GN,CJas}.

\vspace{-.25pc}
\hspace{1pc}
\begin{picture}(0,90)
 \put(20,55){\makebox(0,0){${\rm C}$}}
 \put(35,55){\vector(1,0){20}}
  \put(75,55){\makebox(0,0){${\rm Ext}$}}
 \put(88,60){\vector(2,1){18}}
 \put(120,70){\makebox(0,0){${\rm AC}$}}
 \put(132,70){\vector(1,0){30}}
 \put(180,70){\makebox(0,0){${\rm Conn}$}}
 \put(195,70){\vector(1,0){20}}
 \put(224,70){\makebox(0,0){${{\rm D}}$}}
 \put(233,65){\vector(2,-1){18}}
 \put(270,55){\makebox(0,0){${{\rm PC}}$}}
 \put(127,40){\makebox(0,0){${{\rm SCIVP}}$}}
  \put(180,40){\makebox(0,0){${\rm CIVP}$}}
   \put(225,40){\makebox(0,0){${\rm PR}$}}
\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){${{\rm WCIVP}}$}}
\end{picture}
\vspace{-4.5pc}
\begin{center}
\vspace{3.25pc}
Chart~3: Darboux-like functions from $\real$ into $\real$.
\end{center}
The inclusions
${\rm Ext}\subset{\rm AC}\subset{\rm Conn}$
were proved by J.~Stallings
while the containment ${\rm Ext}\subset{\rm SCIVP}$ was proved
by H.~Rosen, R.G.~Gibson, and F.~Roush.

The main interest in Darboux-like functions comes
from the fact that
the class $\Delta'$ of the derivatives from $\real$ into $\real$
is contained in all these classes, that is,
${\rm C}\subsetneq\Delta'\subsetneq{\rm Ext}$.
This follows from the fact that every derivative is
Darboux Baire class~1  (see e.g. \cite{Br})
while within the Baire class~1 Chart~3 reduces to
\[
{\rm C}\longrightarrow{\rm Ext}=
{\rm AC}={\rm Conn}={\rm D}={\rm PC}={\rm SCIVP}={\rm CIVP}=
{\rm PR}\longrightarrow{\rm WCIVP}.
\]
The proof that every peripherally continuous Baire class~1
function $f\colon\real\to\real$ is extendable
is due to J.~Brown, P.~Humke, and M.~Laczkovich~\cite{BHL}.
In fact, most of the properties used to define Darboux-like functions
were introduced as characterizations of Darboux functions within the Baire
class~1,
in a form ``a~Baire class~1 function $f$
is {\em Darboux}\/ if and only if
$f$ satisfies the {\em given property}.''
But these properties make sense without the Baire class~1 restriction, so it
was natural to study these various conditions on their own, and to
find the interrelations. A number of mathematicians did just that in
the latter part of the 20th century.

It is interesting that within the Baire class~2 Chart~3
has yet another, quite different form~\cite{CJas}:

\vspace{-10pt}
\hspace{-1.5pc}
\begin{picture}(0,90)
 \put(130,55){\makebox(0,0){${\rm C}
 \longrightarrow{\rm Ext}\longrightarrow{\rm AC}\longrightarrow
 {\rm Conn}\longrightarrow{\rm D}\longrightarrow{\rm SCIVP}={\rm CIVP}$}}
 \put(260,60){\vector(2,1){18}}
 \put(312,70){\makebox(0,0){${\rm PR}\longrightarrow{\rm PC}$}}
 \put(308,40){\makebox(0,0){${{\rm WCIVP}}$}}
 \put(260,52){\vector(2,-1){18}}
\end{picture}\vspace{-2pc}
\vspace{-2.5pc}
\begin{center}
\vspace{1.5pc}
Chart~4: Darboux-like Baire class~2 functions from $\real$ into $\real$.
\end{center}
The most involved work in arguing for this chart
is the nonreversability of the inclusions. (See \cite[thm. 1.2]{CJas}.)
Chart~4 remains unchanged if we restrict
Darboux-like functions to \otherterm{Borel functions} in place of Baire
class~2.   Within the class of additive functions Chart~3 remains almost
unchanged: the only difference is that in this case we have
${\rm PR}={\rm WCIVP}$ and that the example of additive
function $f\colon\real\to\real$ from
${\rm Conn}\setminus{\rm AC}$ (which is also ${\rm CIVP}$)
is known only under
an extra set theoretical
assumption that the union of less than continuum
many meager subsets of $\real$ is meager in $\real$.
(A subset of a topological space $X$ is \defterm{meager},
or of the first category, if it is a countable union of
\otherterm{nowhere dense} subsets of $X$.)


The Darboux-like classes of functions are not closed under
arithmetic operations.
(See e.g. surveys~\cite{GN,KCsurv}.) For example, if
$f_0$ is the $\sin(1/x)$ function defined above
and $f_1=f_0+\charf{\{0\}}$, where
$\charf{A}$ is a \otherterm{characteristic function} of $A$,
then both $f_0$ and $f_1$ are Darboux Baire class~1, so they are also
extendable. However, $f_1-f_0=\charf{\{0\}}$ is clearly not even
in ${\rm PC}$. In fact, in 1927 A.~Lindenbaum
noticed that every function $f\colon\real\to\real$ can be
written as a sum of two Darboux functions,
while H.~Fast in~1959 proved that for every family
${\mathcal F}\subset\real^\real$ of cardinality
continuum there is just
one Darboux function
$g\in\real^\real$ such that
$g+F\stackrel{\rm def}{=}\{g+f\colon f\in{\mathcal F}\}$
is a subset of ${\rm D}$. The result of H.~Fast is a generalization of
that of A.~Lindenbaum, since it is easy to see that
$\real^\real={\mathcal F}-{\mathcal F}$ if and only
if for every $f,f'\in\real^\real$
there exists a $g\in\real^\real$ such that $g+f,g+f'\in{\mathcal F}$.
(See~\cite[prop.~4.9]{KCsurv}.)
This led T.~Natkaniec~\cite{N1} to study the
following cardinal operator defined
for every ${\mathcal F}\subset\real^X$, where $|X|$ stands for the
cardinality of $X$:
\[
{\rm A}({\mathcal F}) =
\min\left\{|H|\colon H\subset\real^X\ \&\
\neg\exists g\in\real^X\ g+H\subset{\mathcal F}\right\}
\cup\left\{\left|\real^X\right|^+\right\}.
\]
The values of the operator $A$ for Darboux-like classes
of functions from $\real$ to $\real$ are as follows
(see e.g. \cite[thms.~4.7~\&~4.10]{KCsurv}):
$${\mathfrak c}^+={\rm A}({\rm PR})={\rm A}({\rm Ext})
\leq {\rm A}({\rm AC})={\rm A}({\rm D})
\leq {\rm A}({\rm PC})=2^{\mathfrak c},
$$
where the value of ${\rm A}({\rm D})$ between
${\mathfrak c}^+$ and $2^{\mathfrak c}$ can vary
in different models of ZFC.
Moreover, the monotonicity of the operator $A$
implies that
${\rm A}({\rm Ext})={\rm A}({\rm SCIVP})=
{\rm A}({\rm CIVP})={\rm A}({\rm PR})={\mathfrak c}^+$
and ${\rm A}({\rm AC})={\rm A}({\rm Conn})={\rm A}({\rm D})$.

The above discussion shows that unlike the derivatives,
classes of Darboux-like functions are not closed under
addition. It is also not difficult to see that
none of these classes (including $\Delta'$, see~\cite[p.~14]{Br})
is closed under multiplication.
Closure under composition
gives a completely different picture.
First of all, the derivatives are not closed
under composition: by a theorem of I.~Maximoff
(see e.g.~\cite[p.~26]{Br}), for every Darboux Baire class~1 function
$g\colon\real\to\real$
(which does not need to be a derivative) there
exists a homeomorphism $h$ of $\real$
such that $f=g\circ h$ is a derivative;
so, the composition $g=f\circ h^{-1}$ does not need to be a derivative.
It is obvious from the definition that
the class ${\rm D}$ of Darboux functions
is closed under composition, and clearly so is ${\rm C}$.
The other classes from Chart~3, except for ${\rm Ext}$,
are not closed under composition.
The problem of closure of ${\rm Ext}$ under composition remains
open~\cite[Q.~9.1]{GNup}.
In fact, it is even not known whether
the composition of two derivatives must be in ${\rm Conn}$.
A partial positive result in this direction
was recently obtained by
M.~Cs\"ornyei, T.~C.~O'Neil,
D.~Preiss~\cite{COP} and, independently, by
M.~ Elekes, T.~Keleti, V.~Prokaj~\cite{EKP}, 
who proved that
the composition of two derivatives from $[0,1]$ into $[0,1]$
must have a fixed point. (So, we cannot exclude the possibility
that $\Delta'\circ\Delta'\subset{\rm AC}$.)

The main reason for the studies of classes of functions
related to derivatives comes from the fact
that the class $\Delta'$ of all derivatives does not have
any known nice characterization. (See~\cite{Br}.)
One of the recent attempts of finding 
a characterization was to
\defterm{topologize} it, that is to find two
\otherterm{topologies} $\tau_0$ and $\tau_1$ on $\real$
for which $\Delta'$ is equal to the class
${\rm C}(\tau_0,\tau_1)$ of all continuous functions
from $\langle\real,\tau_0\rangle$ into $\langle\real,\tau_1\rangle$.
Unfortunately, $\Delta'$ cannot be topologized, as
shown by K.~Ciesielski and, independently, by M.~Tartaglia.
(See~\cite[cor.~5.5]{KCsurv}.)
However K.~Ciesielski~\cite{KC:Der}
proved that it can be characterized by preimages of sets
in the sense that there exist families
${\mathcal A}$ and ${\mathcal B}$ of subsets of $\real$
with the property
that
$\Delta'=
\{f\in\real^\real\colon f^{-1}(B)\in{\mathcal A}\
\mbox{ for every }\ B\in{\mathcal B}\}$.
It is interesting, that if the \otherterm{generalized continuum
hypothesis} holds then many classes of functions can be
topologized~\cite[sec.~5]{KCsurv}.
In particular, this is the case for any class ${\mathcal F}$ of functions
containing all constant functions such that
${\mathcal F}$ is  contained either in the class of
analytic functions from $\real$ to $\real$ or in the class
of harmonic functions from $\real^2$ to~$\real^2$.

The class $\Delta'$ is also closely related to the class $\rm Appr$ of
\defterm{approximately continuous functions}, which was
introduced  by A.~Denjoy in 1915.
(See e.g.~\cite{Br,CLO:book}.)
Recall that $f\colon\real\to\real$ is in $\rm Appr$ provided
for every $x_0\in\real$ the
\defterm{approximate limit}
$\operatorname{applim}_{x\to x_0} f(x)$ equals to $f(x_0)$, where
$\operatorname{applim}_{x\to x_0} f(x)=L$ if
there exists a set $S\subset\real$ such that
$x_0$ is a (Lebesgue) \defterm{density point} of $S$
(that is, $\lim_{h\to 0^+}\frac{\lambda(S\cap[x_0-h,x_0+h])}{2h}=1$,
with $\lambda(A)$ standing for the \otherterm{inner Lebesgue measure} of $A$)
and
$\lim_{x\to x_0,\, x\in S}f(x)=L$.
The interest in $\rm Appr$ comes from the fact that
every bounded approximately continuous functions is a derivative.
Also, each function in $\rm Appr$
is Darboux Baire class~1, so it belongs to every class 
of  Darboux-like functions. It was not until 1952 that
O.~Haupt and C.~Pauc
defined the \defterm{density topology} $\tau_{\mathcal N}$
on $\real$ (which is the family of all $G\subset\real$ such that
every $x\in G$ is a density point of $G$)
and showed that
$\rm Appr$ is equal to the class ${\rm C}(\tau_{\mathcal N},\tau_O)$
of all functions continuous with respect to the
density topology $\tau_{\mathcal N}$
on the domain and the ordinary topology $\tau_O$ on the range.
Their paper seemed to have had almost no
impact and the same results were rediscovered in 1961 by
C.~Goffman and D.~Waterman. (See~\cite[sec.~1.5]{CLO:book}.)
This led to deep studies of the density topology,
as well as to its category analog $\tau_{\mathcal I}$,
known under the name
of \defterm{${\mathcal I}$-density topology}.
Extensive research have been also conducted on the classes
${\rm C}(\tau_{\mathcal I},\tau_O)$ of
\defterm{${\mathcal I}$-approximately continuous functions},
${\rm C}(\tau_{\mathcal N},\tau_{\mathcal N})$ of
\defterm{density continuous functions}, and
${\rm C}(\tau_{\mathcal I},\tau_{\mathcal I})$ of
\defterm{${\mathcal I}$-density continuous functions}.
(See~\cite{CLO:book}.)
Classes ${\rm C}(\tau_{\mathcal N},\tau_O)$
and ${\rm C}(\tau_{\mathcal I},\tau_O)$
are closed under addition, while
${\rm C}(\tau_{\mathcal N},\tau_{\mathcal N})$ and
${\rm C}(\tau_{\mathcal I},\tau_{\mathcal I})$ are
not.

A function $f\colon\real\to\real$ is
\defterm{symmetrically continuous}, $f\in{\rm SC}$, provided
$\lim_{h\to 0}\left[f(x+h)-f(x-h)\right] =0$ for every
$x\in\real$; it is
\defterm{approximately symmetrically continuous},
$f\in{\rm ApprSC}$, if
$\operatorname{applim}_{h\to 0}\left[f(x+h)-f(x-h)\right] =0$ for each
$x\in\real$. The theory of symmetrically continuous functions
stems from the theory of trigonometric series and
dates back to the beginning of the 20th century.
Rresearch in this area has been very active in the last several
years (see~\cite{Thomson}) after C.~Freiling and D.~Rinne
in 1988 solved a long standing problem proving
that every measurable function $f\colon\real\to\real$
having \defterm{approximate symmetric derivative},
$D^s_{ap}f(x)\stackrel{\rm def}{=}
\operatorname{applim}_{h\to 0}\frac{f(x+h)-f(x-h)}{2h}$,
equal $0$ for all $x\in\real$ must be constant almost everywhere.
C.~Freiling~\cite{Fr} also proved  that it is consistent with
ZFC that in the above theorem the assumption of a measurability
of $f$ can be dropped. However, this cannot be proved in ZFC,
as under the \otherterm{continuum hypothesis},
CH, W.~Sierpi\'nski constructed a nonempty
proper subset $X$ of $\real$
for which $f=\charf{X}$ has approximate symmetric derivative
equal $0$ for all $x\in\real$.
The above classes can be added to Chart~3 as follows.


\hspace{2pc}
\begin{picture}(0,90)
 \put(75,55){\makebox(0,0){${\rm C}$}}
    \put(88,60){\vector(2,1){18}}
       \put(122,70){\makebox(0,0){$\rm Appr$}}
       \put(140,70){\vector(4,1){20}}
       \put(175,77){\makebox(0,0){${\rm Ext}$}}
    \put(140,65){\vector(2,-1){18}}
    \put(184,55){\makebox(0,0){$\rm ApprSC$}}
    \put(88,52){\vector(2,-1){18}}
       \put(124,40){\makebox(0,0){${\rm SC}$}}
       \put(140,40){\vector(4,-1){20}}
       \put(190,34){\makebox(0,0){$\rm Measurable$}}
    \put(140,43){\vector(2,1){18}}
\end{picture}
\vspace{-3pc}
\begin{center}
Chart~5: Approximately and symmetrically continuous functions.
\end{center}
The fact that every
symmetrically continuous function
is measurable follows from
a theorem
I.N.~Pesin and D.~Preiss~\cite[thm.~2.26]{Thomson} asserting
that if an $f$ is symmetrically continuous
then its set of points of discontinuity is meager and of measure zero.
On the other hand, an approximately symmetrically continuous
function need not  be measurable, as
witnessed by the function $f=\charf{X}$ mentioned above.
It is unknown whether
there exists a ZFC example of a nonmeasurable function in $\rm ApprSC$.

It is easy to see that all inclusions in Chart~5 are proper.
For example,
clearly $\charf{\{0\}}\in{\rm SC}\setminus{\rm Appr}$.
To get $g\in{\rm Appr}\setminus{\rm SC}$ take
$a_0<b_0<a_1<b_1<\cdots<0$ such that
$0$ is an accumulation point of $E=\bigcup_{n<\omega}[a_n,b_n]$
and a density point of $\real\setminus E$. Put
$g(x)=0$ for $x\in\real\setminus E$ and
$g(x)=(b_n-a_n)^{-1}{\rm dist}(x,\real\setminus[a_n,b_n])$
for $x\in[a_n,b_n]$.
Finally $g+\charf{\{0\}}\in{\rm ApprSC}\setminus({\rm Appr}\cup{\rm SC})$.

Another interesting notion of generalized continuity is known as
\defterm{countable continuity}.
For topological spaces
$X$, $Y$ and ${\mathcal F}\subset Y^X$, we define
\defterm{dec$({\mathcal F})$} as the smallest infinite cardinal
$\kappa$ such that for every $f\in{\mathcal F}$ there is
a family of at most $\kappa$-many continuous functions $g$
from a subset of $X$ into $Y$ such $f$ is covered by
all these functions $g$. A function $f\colon X\to Y$ is
\defterm{$\kappa$-continuous} provided ${\rm dec}(\{f\})\leq\kappa$;
it is \defterm{countably continuous} if $f$ is
$\omega$-continuous. (See~\cite[sec.~4]{KCsurv}.)
The study of these notions
was initiated by a question of N.~Luzin
whether every Borel function from $\real$ into $\real$
is countably continuous.
This question was answered negatively by P.S.~Novikov
and generalized by L.~Keldy\v{s}.
In fact we have already ${\rm dec}({\mathcal B}_1)>\omega$, where
${\mathcal B}_1$ is the family of Baire class~1 functions
from $\real$ to $\real$.
The most general result in this direction was
obtained by
J.~Cicho\'{n}, M.~Morayne, J.~Pawlikowski,
and S.~Solecki~\cite[thm.~4.1]{KCsurv}
who proved that
${\rm cov}({\mathcal M})\leq {\rm dec}({\mathcal B}_1)\leq d$,
where ${\rm cov}({\mathcal M})$ is the smallest cardinality of
a \otherterm{covering} of $\real$
by meager sets, and $d$
is the \otherterm{dominating number}.
The consistency of ${\rm cov}({\mathcal M})<{\rm dec}({\mathcal B}_1)$
and
${\rm dec}({\mathcal B}_1)<d$ was proved by
S.~Stepr\={a}ns~\cite[thm.~4.2]{KCsurv},
and S.~Shelah, S.~Stepr\={a}ns~\cite[thm.~4.3]{KCsurv},
respectively.
Number $\rm dec$ has been also studied by
K.~Ciesielski in~\cite{62:DecSymmContv}, 
in which he proved that
${\rm cof}({\mathfrak c})\leq{\rm dec}({\rm SC})
={\rm dec}({\rm SZ})={\rm dec}(\real^\real)\leq{\mathfrak c}$
and that each of the inequalities
can be strict. ${\rm cof}({\mathfrak c})$
stands for the \otherterm{cofinality} of the continuum ${\mathfrak c}$
and ${\rm SZ}$ for the class of \defterm{Sierpi\'nski-Zygmund
functions} $f\colon\real\to\real$, that is,
those whose restriction
$f\restriction X$ is discontinuous for every $X\subset\real$
of cardinality~${\mathfrak c}$.

Finally, one can ask how much continuity
an arbitrary function from
a topological space $X$ into $Y$ must have. (See e.g. \cite[pp.
148-149]{KCsurv}.)
In 1922 H.~Blumberg proved that for every $f\colon\real\to\real$ there
exists a dense subset $D$ of $\real$ such that
$f\restriction D$ is continuous.
This result was generalized by several authors to
more general topological spaces. However, the most interesting discussion
of Blumberg theorem remains in the case of functions
from $\real$ to $\real$.
Blumberg's set $D$ is countable and in ZFC this is
the best that can be proved, since under CH
a restriction of a Sierpi\'nski-Zygmund function
to any uncountable set is discontinuous.
A similar example can be also found in some models of ZFC (e.g. a
\otherterm{Cohen
model})  in the absence of CH as noticed by several authors.
(See~\cite[thm.~2.9]{KCsurv}.) At the same time
S.~Baldwin~\cite[thm.~2.8]{KCsurv}
showed that under \otherterm{Martin's Axiom} for
every function $f\colon\real\to\real$ and every cardinal number
$\kappa<{\mathfrak c}$  there exists a set $D\subset\real$ such that
$f\restriction D$ is continuous and $D$ is \defterm{$\kappa$-dense},
that is,
$D\cap I$ has cardinality at least $\kappa$ for every
nondegenerated interval $I$.
In the same direction S.~Shelah~~\cite[thm.~2.10]{KCsurv}
showed that it is consistent with ZFC
that  for every function $f\colon\real\to\real$
there exists a set $D\subset\real$ such that
$f\restriction D$ is continuous and $D$ is \defterm{nowhere meager},
that is,
$D\cap I$ is nonmeager for every nontrivial interval $I$.
Most recently,  A.~Ros{\l}anowski and S.~Shelah (unpublished)
also found a model of ZFC in which it is always possible to find
the set $D$ of positive \otherterm{outer measure},
though in this case we cannot require that $D$ is dense in~$\real$.
(See~\cite[thm.~2.11]{KCsurv}.)

Its is easy to find a function $f\colon\real\to\real$ which
has no points of continuity---the characteristic function of
the set of rational numbers has this property.
But what if we ask for points of continuity
in weaker sense? For example, 
a function $f\colon\real\to\real$
is \defterm{weakly continuous} at
$x$ if it has the Young property at $x$, that is, if there
are sequences $a_n\nearrow 0$ and $b_n\searrow 0$ 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.
More precisely,
every $f\colon\real\to\real$
is weakly continuous everywhere on the complement of a
countable set. (See~\cite[thm.~2.16]{KCsurv}.)
A natural symmetric counterpart of weak continuity is
defined as follows: a
function $f\colon\real\to\real$ is
\defterm{weakly symmetrically continuous} at $x$
if there is a sequence $h_n\to 0$ such that
$\lim_{n\to\infty}(f(x+h_n)-f(x-h_n))=0$.
The symmetric version of the theorem mentioned above
badly fails: K.~Ciesielski and L.~Larson~\cite[thm.~2.17]{KCsurv}
constructed a nowhere weakly symmetrically continuous functions
$f\colon\real\to\{0,1,2,3,\ldots\}$.
It is unknown whether
a nowhere weakly symmetrically continuous functions
$f\colon\real\to\real$
can have finite range~\cite[prob.~2]{KCsurv},
though its range must have at least four elements.
K.~Ciesielski and S.~Shelah~\cite{CS}
proved that such an $f$ can have bounded countable range.

For functions from $\real$ to $\real$
many generalized continuities
mentioned above can be viewed in the context of \defterm{path limit}
$P$-$\lim_{x\to x_0}f(x)\stackrel{\rm def}{=}\lim_{x\to x_0,\, x\in P}f(x)$
where $x_0$ is in the closure of $P\cap(x_0,\infty)$
and of $P\cap(-\infty,x_0)$.
Thus,
for a continuous function the path $P$ at $x_0$ must be an interval;
for $f\in{\rm PR}$ a path must be a perfect set;
for $f\in{\rm PC}$ (i.e., weakly continuous) any path $P$ works;
for an approximately
continuous function $x_0$ must be a density point of a path $P$;
in any symmetric version of these notions the paths must be symmetric with
respect to $x_0$.
Luzin's theorem implies that every bounded measurable function $f\colon\real\to\real$
is approximately continuous almost everywhere. Blumberg's theorem
and Sierpi\'nski-Zygmund's
example illustrate the extent to which arbitrary functions have sets
of restricted continuity.



Certainly, the above discussion barely touches
the tip of the iceberg of different notions
of generalized continuities. From the notions
not mentioned so far probably the most studied
is that of quasi-continuity
introduced in 1932 by Kempisty. (See \cite[sec.~6]{GN}.) Thus,
a function $f$ from a topological space $X$ into $\real$
is \defterm{quasi-continuous},
$f\in {\rm QC}(X)$, if for every $x\in X$
and open sets $U\ni x$ and $V\ni f(x)$ there exists
a nonempty open $W\subset U$ with $f[W]\subset V$.
The other two closely related classes
are defined as follows.
A function $f\colon X\to\real$
is \defterm{cliquish}, $f\in {\rm CLIQ}(X)$,
if for every $x\in X$, open
$U\ni x$, and $\varepsilon>0$ there
is a nonempty open $W\subset U$ such that
$|f(y)-f(z)|<\varepsilon$ for all $y,z\in W$;
$f$ is \defterm{almost continuous in sense of Husain},
$f\in {\rm ACH}(X)$,
if for every $x\in X$ and open $V\ni f(x)$
point $x$ belongs to the interior of the closure of
$f^{-1}(V)$.
It is not difficult to see that
${\rm QC}(X)\subset{\rm CLIQ}(X)$
and that every $f\in {\rm CLIQ}(X)$ has the
\otherterm{Baire property}.
Quasi-continuous functions need not to be in ${\rm PC}$,
as witnessed by $\charf{(0,\infty)}$.
Also, ${\rm Ext}\not\subset {\rm CLIQ}(\real)$, since
${\rm Ext}+{\rm Ext}=\real^\real\neq {\rm CLIQ}(\real)=
{\rm CLIQ}(\real)+{\rm CLIQ}(\real)$.
The relation ${\rm Ext}\not\subset{\rm ACH}(\real)\not\subset{\rm D}$
is justified by a $\sin(1/x)$-function
and the characteristic function of
the set of rational numbers, respectively.
However, we have ${\rm ACH}(\real)\subset{\rm PC}$.
Also, ${\rm QC}(\real)\not\subset{\rm ACH}(\real)$
and ${\rm ACH}(\real)\not\subset{\rm CLIQ}(\real)$,
where the second relation is justified by the
characteristic function of a
\otherterm{Bernstein set}.




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\end{document}
