\documentclass{rae}

\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}


\MathReviews{Primary 26A15; Secondary 54C30.}

\keywords{Darboux functions, extendable functions, almost
continuous functions, connectivity functions, functions with
 perfect road, peripherally continuous functions,
CIVP-functions, SCIVP-functions, quasi-continuous functions}



\markboth{R. G. Gibson and T. Natkaniec}{Darboux Like Functions}

\author{Richard G. Gibson, Department of Mathematics,
Columbus State University, Columbus, Georgia, 31907, USA,
email: gibson\_richard@colstate.edu\\
Tomasz Natkaniec\footnote{This work
was partially supported by NSF Cooperative
Research Grant INT-9600548 with its Polish part
financed by KBN. The final version of the paper was prepared during
a visit of the second author at Columbus State University and West
Virginia University.}, Department of Mathematics, Gda{\'n}sk
 University, Wita Stwosza 57, 80-952 Gda{\'n}sk, Poland,
email: mattn@ksinet.univ.gda.pl}

\title{DARBOUX LIKE FUNCTIONS. OLD~PROBLEMS~AND~NEW~RESULTS}


\newcommand{\F}{{\cal F}}
\def\P{{\cal P}}
\newcommand{\A}{{\cal A}}
\newcommand{\B}{{\cal B}}
\newcommand{\C}{{\cal C}}
\newcommand{\K}{{\cal K}}
\newcommand{\CAB}{{\C_{\A,\B}}}
\newcommand{\DAB}{{\C^{-1}_{\A,\B}}}
\newcommand{\ACS}{{\rm ACS}}
\newcommand{\Const}{{\rm Const}}
\newcommand{\G}{{\cal G}}
\newcommand{\CC}{{\rm C}}
\newcommand{\PR}{{\rm PR}}
\newcommand{\PC}{{\rm PC}}
\newcommand{\PB}{{\rm PB}}
\newcommand{\PI}{{\rm PI}}
\newcommand{\D}{{\rm D}}
\newcommand{\Conn}{{\rm Conn}}
\newcommand{\CIVP}{{\rm CIVP}}
\newcommand{\SCIVP}{{\rm SCIVP}}
\newcommand{\WCIVP}{{\rm WCIVP}}
\newcommand{\DIVP}{{\rm DIVP}}
\newcommand{\QU}{{\rm QU}}
\newcommand{\Ext}{{\rm Ext}}
\newcommand{\Q}{{\rm Q}}


\newtheorem{Th}{Theorem}
\newtheorem{Le}[Th]{Lemma}
\newtheorem{Co}[Th]{Corollary}
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\newtheorem{Qu}[Th]{Question}
\newtheorem{Pro}[Th]{Problem}
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\newcommand\real{\mathR}
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\newcommand{\restr}{{\restriction}}
\newcommand{\co}{{\mathfrak c}}

\newcommand\id{{\rm id}}
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\newcommand{\pf}{{\noindent\sc Proof. }}
\newcommand{\charf}[1]{\mbox{\raise.48ex\hbox{$\chi$}$_{#1}$}}

\begin{document}
\maketitle

During the 14th Summer School on Real Functions Theory
in Liptovsky J{\'a}n, Slovakia in 1996, Richard G. (Jerry) Gibson
gave a talk
concerning Darboux-like functions. Following his talk we wrote the
survey
article {\it Darboux-like functions}~\cite{GN}, that was published in
the
{\it Real Analysis Exchange}. In that paper we collected not only
known facts on the subject but also presented some open problems. In
the last
two years a considerable numbers of papers have been written that
solve
some of the
open questions posed in our survey article. In this short abstract we
would
like
to present those solutions and to list the problems that remain open.
It is
based
on the talk given by Tomasz (Tomek) Natkaniec at the 15th Summer
School
on Real Functions Theory in Liptovsky J{\'a}n, Slovakia, in 1998.

Most of the new results are contained as a part of the Polish-American
project {\it Set Theoretic Analysis}
and has been described in K.~Ciesielski's survey
{\it Set Theoretic Real Analysis}~\cite{C}.
You can find these results on the Set
Theoretic Analysis web page:

\centerline{http://www.math.wvu.edu/homepages/kcies/STA/STA.html}

\section{Notations and definitions}
We use the notations from our survey article~\cite{GN}. In
particular, the
questions in this paper are numbered the same as in the survey
article.

Recall the main notations and definitions from the survey article.

\begin{description}
\item[D]
 -- $f$ is a {\it Darboux function} if $f(C)$ is connected
 whenever $C$ is connected in $X$;
\item[Conn]
 -- $f$ is a {\it connectivity function} if the graph of $f$
 restricted to $C$ is connected in
 $X\times Y$ whenever $C\subset X$ is connected;
\item[ACS]
 -- $f$ is an {\it almost continuous function} in the sense of
 Stallings, if $U$ is an open subset of $X\times Y$ containing
 the graph of $f$, then $U$ contains the graph of a continuous
 function $g:X\to Y$;
\item[Ext]
 -- $f$ is an {\it extendable function} if there exists a
 connectivity function $g:X\times I\to Y$ such that $f(x)=g(x,0)$ for
all $x\in X$.
\end{description}

\section{Borel measurable functions}

\medskip
It is well known that in the class of all functions from $\mathR$ into
$\mathR$ the following chain of implications holds:
$$
(\star)\;\;\;\Ext \rightarrow\ACS \rightarrow\Conn \rightarrow\D
$$
For Baire class 1 functions, $f\colon\mathR\to\mathR$, the properties
defined above are equivalent, i.e.,
$$\Ext=\ACS=\Conn=\D.$$
However, for Borel measurable functions, Brown, Humke and Laczkovich
 proved that the implications $(\star)$  are not reversible except for possibly
 $\Ext \rightarrow\ACS$~\cite{BHL}.
Thus they posed the following question.
\medskip

\noindent
{\bf Question~3.21. }{\em Does there exist a Borel function
$f\colon\mathR\to
\mathR$ such that $f\in\ACS\setminus\Ext$?}
\medskip

The answer to this question was given in the following theorem.
\medskip
\begin{Th}\label{th1}
{\rm (Ciesielski,  Jastrz{\c{e}}bski, 1998) \cite{CJ}}
$$(\ACS\setminus \Ext)\cap B_2\neq \emptyset.$$
\end{Th}
This answers also {\bf Question~3.11}. (See the next section.)

\medskip
J.~Brown~\cite{JB} has considered whether the implications
$(\star)$ within the class of all functions with $G_{\delta}$
graphs are reversible. He noticed that in this class all
implications $\ACS \rightarrow\Conn \rightarrow\D$ are not
reversible. (Recall that every function in the class $B_1$ has a
graph which is a $G_{\delta}$ subset of $\mathR^2$. Moreover, it is
well-known that every
 function with a $G_{\delta}$ graph is Borel measurable. However, it can be
of an arbitrary high Borel class. See e.g., \cite{AK}.)

The problem of whether the implication $\Ext\rightarrow\ACS$ in the class
of all functions with a $G_{\delta}$ graph is reversible
 was posed by R.~Gibson~\cite{RG}.

\medskip
\noindent
{\bf Question~3.15. }{\em Does there exist a function
$f\in\ACS\setminus\Ext$
with a $G_{\delta}$ graph?}

\medskip

This question was answered in the next theorem.
\medskip

\begin{Th}
{\rm (Ciesielski, Rosen, 1999) \cite{CRos} There exists a Baire two function
$f\in\ACS\setminus\Ext$ with a $G_{\delta}$ graph.}
\end{Th}

The next problem is connected with the
\underline{Ces{\'a}ro-Vietoris function} $\varphi:{\bf I} \to {\bf
I}$ defined by the formula:
$$\varphi(x)= \overline{\lim}_{n
\to\infty}\frac{a_{1}+ \ldots +a_{n}}{n}$$
 where $a_{i}$ are given by the unique nonterminating binary
 expansion of the number $x=(0.a_{1}a_{2} \ldots )$.

\medskip
\noindent
{\bf Claims. }
\begin{itemize}
\item
$\varphi\in B_2$.
\item
{\em (Vietoris, 1921). $\varphi\in\Conn$.}
\item
{\em (Brown, 1975). $\varphi\in\ACS$.}
\end{itemize}

\medskip
\noindent
{\bf Question~3.22. }{\em Does the Ces{\'a}ro-Vietoris function
belong to
$\Ext$?}

{\bf Open.}

\section{Cantor intermediate values  properties}

Recall the following definitions.

\begin{description}
\item[PR]
 -- $f$ has a {\it perfect road} if for every $x\in \mathR$,
 there exists a perfect set $P$ having $x$ as a bilateral limit
 point such that $f\restr P$ is continuous at $x$;
\item[CIVP]
 -- {\it Cantor Intermediate Value Property}: $f\in\CIVP$ if for
 all $p,q\in \mathR$ with $p\neq q$ and $f(p)\neq f(q)$ and for
 every Cantor set $K$ between $f(p)$ and $f(q)$, there exists a
 Cantor set $C$ between $p$ and $q$ such that $f(C)\subset
 K$;
\item[SCIVP]
 -- {\it Strong Cantor Intermediate Value Property}: $f\in\SCIVP$
 if for all $p,q\in\mathR$ with $p\neq q$ and $f(p)\neq f(q)$ and
 for every Cantor set $K$ between $f(p)$ and $f(q)$, there exists
 a Cantor set $C$ between $p$ and $q$ such that $f(C)\subset K$ and
$f\restr C$ is continuous.

\item[PC]
-- $f$ is {\it peripherally continuous} 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 $W\subset U$
such that $x\in W$ and $f(\bd(W))\subset V$, where $\bd(W)$ denotes
the boundary of $W$.

\end{description}

\medskip
For Baire class 1 functions, $f\colon\mathR\to\mathR$, the functions
defined above are equivalent, i.e.,

$$\Ext=\ACS=\Conn=\D=\SCIVP=\CIVP=\PR=\PC.$$

\medskip
For arbitrary function, $f\colon\mathR\to\mathR$, we have only the
following implications

\begin{picture}(0,95)
\put(30,50){\makebox(0,0){$\Ext$}} \put(40,52){\vector(1,1){20}}
\put(40,48){\vector(1,-1){20}} \put(80,72){\makebox(0,0){$\ACS$}}
\put(85,30){\makebox(0,0){$\SCIVP$}}
 \put(95,72){\vector(1,0){35}}
 \put(148,72){\makebox(0,0){$\Conn$}}
  \put(165,72){\vector(1,0){35}}
\put(210,72){\makebox(0,0){$\D$}} \put(110,30){\vector(1,0){20}}
\put(155,30){\makebox(0,0){$\CIVP$}}
\put(175,30){\vector(1,0){20}} \put(210,30){\makebox(0,0){$\PR$}}
 \put(225,30){\vector(1,1){20}}
\put(225,72){\vector(1,-1){20}}
 \put(255,50){\makebox(0,0){$\PC$}}
\end{picture}
\begin{center}
Chart 1
\end{center}

\medskip
The following theorems and questions are related to Chart 1.
\medskip

\begin{Th}
{\rm (K. Banaszewski, Natkaniec, 1996)~\cite{BN}}
Assume CH. Then
$(\ACS\cap\CIVP)\setminus\Ext\neq\emptyset$.
\end{Th}

\medskip
\noindent
{\bf Question. }{\em Can the above be proved in ZFC?}

\begin{Th}
{\rm (Ciesielski, 1997)~\cite{KC}} Yes.
\end{Th}

\medskip
\noindent
{\bf Question~3.11. }{\em Does there exist $f\in
(\ACS\cap\SCIVP)\setminus\Ext?$}

\begin{Th}
{\rm (Rosen, 1997)~\cite{HR}}  Yes, under CH.
\end{Th}

\begin{Th}
{\rm (Ciesielski, Ros{\l}anowski, 1998)~\cite{CR}}
Every $g\in\Ext$ with a dense graph satisfies the following condition:
\begin{description}
\item[SSCIVP]
-- for all $a<b$ and for every perfect set $K$ between $g(a)$ and
$g(b)$ there exists a perfect set $C\subset (a,b)$ such that
$g(C)\subset K$ and $g|C$ is continuous strictly
increasing.\footnote{``{\it strictly increasing}'' in this statement can be
replaced by ``{\it strictly decreasing}''}
\end{description}
\end{Th}
(Note that there exists an $f\in (\ACS\cap {\rm SSCIVP})\setminus\Ext$.
This example has been constructed recently by K.~Ciesielski.\footnote{Unpublished.})

The following corollary follows immediately from
the above theorem, as well as from Theorem~\ref{th1}.\footnote{This
is a consequence of the fact that every Borel measurable Darboux
function has the $\SCIVP$.}
\begin{Co}{\rm \cite{CR}, \cite{CJ}}
$(\ACS\cap\SCIVP)\setminus\Ext\neq\emptyset$.
\end{Co}

\section{Additive functions}

Recall that a function $f\colon\mathR\to\mathR$ is additive if it
satisfies the Cauchy equation: $f(x+y)=f(x)+f(y)$ for each
$x,y\in\mathR$.

\medskip
\noindent
{\bf Question~5.2. }{\em Does there exist a discontinuous additive
$\ACS$  function whose graph is small in the sense of measure or
category?}

\begin{Th}
{\rm (Ciesielski, 1997)~\cite{KC}}
\begin{itemize}
\item
If $\mathR$ is not a union of less than continuum meager
sets then there exists an additive discontinuous $\ACS$ function
 $f$ with the graph of measure zero.
\item
If $\mathR$ is not a union of less than continuum sets of measure
zero then there exists an additive discontinuous $\ACS$ function $f$
 with a meager graph.
\item
Under CH there exists an additive
discontinuous $\ACS$ function $f$ with the graph which is meager and
of
measure zero.\footnote{J.~Pawlikowski has recently noticed that
the same example exists also under the Martin's axiom.}
\end{itemize}
\end{Th}

{\bf Question 5.2 remains open in ZFC.}

\medskip
\noindent
{\bf Question~5.5. }{\em Does there exist
$f\in {\rm Add}\cap\Conn\setminus \ACS?$}

\begin{Th}
{\rm (Ciesielski, Ros{\l}anowski, 1998)~\cite{CR}} Yes, under CH.
\end{Th}

{\bf Open in ZFC.}

\medskip
\noindent
{\bf Question. }{\em Does there exist a discontinuous function
$f\colon\mathR^n\to\mathR$, $f\in {\rm Add}\cap\Ext?$}

\begin{Th}
{\rm (Ciesielski, Jastrz{\c{e}}bski, 1998)~\cite{CJ}} Yes, for $n=1$.
No for $n>1$.
\end{Th}

\section{Darboux like functions and quasi-continuity}

In general Darboux like functions and quasi-continuous functions are
not related.
However, under certain conditions quasi-continuous functions are
extendable.
One of these conditions is that the function have a graph whose closure
is bilaterally dense in itself. (It means that the right and left cluster sets
of $f$ at any point $x$ of the domain of $f$ coincide.)

\begin{description}
\item[QC]
--$f:X\to Y$ is a quasi-continuous function if and only if for each
$p\in X$ the following condition holds: for every open set
$U\subset X$ with $p\in U$ and open set $V\subset Y$ with $f(p)\in V$
there exist a non-empty open set $W\subset U$ such that
$f(W)\subset V$.
\end{description}

\begin{Th}
{\rm (Rosen, 1998)~\cite{HR2}} If $f$ is Darboux, quasi-continuous and
has a
graph whose closure is bilaterally dense in itself, then $f$ is
extendable and $D(f)$ is $f$-negligible.
\end{Th}

(Here $D(f)$ denotes the set of all points at which $f$ is discontinuous.)

Recently, Francis Jordan \cite{FJ} improved this results by showing that
we need
not assume that the function is a Darboux function.

\section{Functions of $n>1$ variables}

For $n>1$, the implications in Chart 1 are no longer valid.
In fact, we have the followig diagram.

\begin{picture}(0,95)
\put(60,50){\makebox(0,0){$\Ext$}}
\put(80,51){\vector(1,0){20}}
\put(135,50){\makebox(0,0){$\Conn=\PC$}}
\put(170,52){\vector(1,1){20}}
\put(170,48){\vector(1,-1){20}}
\put(210,72){\makebox(0,0){$\ACS$}}
\put(210,30){\makebox(0,0){$\D$}}
\end{picture}

\medskip
\noindent
{\bf Question~8.3. }{\em Is the inclusion $\Ext\subset\Conn$ proper
for
$f\colon\mathR^n\to\mathR$, if $n>1$?}

\begin{Th}
{\rm (Ciesielski, Natkaniec, Wojciechowski, 1998)~\cite{CNW}}  Every
connectivity function $f\colon\mathR^n\to\mathR$, $n>1$ can be
extended to a connectivity function of $n+1$ variables.
\end{Th}

Thus for functions of $n>1$ variables we have the equalities
$$\Ext=\Conn=\PC.$$


\section{Compositions}

\medskip
This section contains the new results concerning compositions of
Darboux like functions.

\medskip
\noindent
{\bf Question~9.3. }{\em Is every Darboux function the composition of
two
(finitely many) $\ACS$ (or $\Conn$) functions?}

\begin{Th}
{\rm (Natkaniec, 1991)~\cite{TN}}
Assume CH.  Every $f\in\D$ with dense level sets is the composition of
two $\ACS$ functions.
\end{Th}

\medskip
\noindent
{\bf Question. }{\em Can the above be proved in ZFC?}

{\bf Open.}

\begin{Th}
{\rm (Kellum, 1998)~\cite{KK}} There exists an $f\in\Conn$ such that
$f$ is not  the composition of finitely many $\ACS$ functions.
\end{Th}

\begin{Th}
{\rm (Ciesielski, Kellum, 1998)~\cite{CK}} There exists an $f\in\D$
such that
$f$ is not the composition of finitely many $\Conn$
functions.
\end{Th}

\noindent
{\bf Question~9.1. }{\em If $f,g\in\Ext$, is $g\circ f\in\Ext$?}

{\bf Open.}

\medskip
The next result seems to be connected with the above problem. (Note
that every $B_2$ function can be expressed as the composition of two
$B_1$ functions.\footnote{This fact was proved for us by S.~Solecki.})

\medskip
\noindent
{\bf Question. }{\em Is every $f\in\D B_2$ the composition of two
derivatives
($\D B_1$ functions)?}

{\bf No. }See Kellum's example in~\cite{KK}. So the folowing problems
seems
to be {\bf open}.

\medskip
\noindent
{\bf Question.} (Ciesielski) {\rm Has the compositions of two derivatives ($\D B_1$ or $\Ext$)
 functions from $\mathI$ into $\mathI$ a fix point?}

(Recall that every connectivity function from $\mathI$ into $\mathI$ (so also every derivative,
every $\D B_1$ and $\Ext$ function) has fixed points.)


\medskip
\noindent
{\bf Question. }{\em Characterize compositions of of two derivatives
($\D B_1$
functions).}

\section{Uniform limits}
Let $\overline{\F}$ denote the class of uniform limits of sequences
of functions from $\F$.

\medskip
\noindent
{\bf Question 9.13. }{\em Does there exists
$f\in\ACS\cap\PR\setminus\overline{\Ext}$?}

\medskip
The answer is affirmative. Both Rosen's function from~\cite{HR}
(constructed under CH) and ZFC Ciesielski-Jastrz{\c{e}}bski's example
from~\cite{CJ} are in the class $\ACS\cap\SCIVP$ but are not in
$\overline{\Ext}$.

\medskip
\noindent
{\bf Question 9.14. }{\em Characterize the uniform limits of
sequences of $\Ext$ functions ($\ACS$ functions or $\Conn$
functions).}

\medskip
This problem is still {\bf open}. However, the following conjecture was formulated by
 K.~Kellum during the last Miniconference in Real Analysis (March 1999) at
 Auburn University.

\medskip
\noindent
{\bf Conjecture.}
{\em Let $f\colon\mathR\to\mathR$ be bounded.
Then $f\in\overline{\rm ACS}$ iff
\begin{itemize}
\item
$f\in {\rm U}\; (=\overline{\rm D})$;
\item
$f$ is away-almost continuous:
\end{itemize}}

A function $f\colon\mathR\to\mathR$ is {\it away almost
continuous} if for every closed set $B\subset\mathR^2$ such that
$\sup\{ {\rm dist}(f(x),B_x)\colon \; x\in \mathR\}>0$ there
exists a continuous function $g\colon\mathR\to\mathR$ which is
disjoint with $B$.


\section{Universal summands}

It is easy to see that there is no $g\in B_{\alpha}$ such that
$B_{\alpha}+g\subset\D$. Similarly, there is no Borel measurable
function $g$ such that $f+g\in\D$ for all Borel functions $f$.

\medskip
\noindent
{\bf Question.}{ \em  {\rm (Ceder, 1965) \cite{JC}}
Does there exist a Borel measurable $g$ such
that $B_{\alpha}+g\subset {\rm D}$?}

\begin{Th}
{\rm (Natkaniec, Rec{\l}aw, 1997)~\cite{NR}} For each $\alpha <
\omega_1$  there is a Borel function with
$B_{\alpha}+g\subset {\rm ACS}$.
\end{Th}

\medskip
\noindent
{\bf Question. }{\em Does there exist $g\in B_{\alpha+1}$ such that
$B_{\alpha}+g\in {\rm D}$ ($B_{\alpha}+g\in {\rm ACS}$ or
$B_{\alpha}+g\in {\rm Ext}$)?}

\begin{Th}
{\rm (Solecki, 1998)~\cite{SS}} For each $\alpha<\omega_1$ there is
a $g\in  B_{\alpha}$ such that $\bigcup_{\beta<\alpha}B_{\beta}+g\in
\Ext$.
\end{Th}

For an ordinal $\alpha<\omega_1$ define the cardinal
$$u(\alpha)=\min(\{ |{\cal F}|\colon\; {\cal F}\subset B_{\alpha}\;\&\; (\forall\; g\in
B_{\alpha})\; ({\cal F}+g\not\subset \D)\}\cup\{ \co^+\})$$
Clearly, $u(0)=\co^+$. Moreover, $u(1)=\omega$~\cite{PP} and $\omega<u(\alpha)<\co$
for $\alpha>1$ (an easy corollary from \cite{JC}). Thus we have the following

\begin{Th}
Under CH, $u(\alpha)=\omega_1$ for all $\alpha>1$.
\end{Th}

\medskip
\noindent
{\bf Question. }{\em Can the above be proved in ZFC?}

{\bf Open.}\footnote{In fact, this is a reformulation of Ceder's problem from~\cite{JC}.}

\section{Products}

\begin{Th}
For every $f\colon\mathR\to\mathR$ the following conditions
are equivalent:
\begin{itemize}
\item
$f$ is a product of two $\D$ functions;
\item
$f$ is a product of two $\Conn$ functions;
\item
$f$ is a product of two $\ACS$ functions;
\item
$f$ possesses the (JC) property:
\begin{quote}
$f$ has a zero in each subinterval in which it changes sign.
\end{quote}
\end{itemize}
\end{Th}

\medskip
\noindent
{\bf Question~9.31. }{\em Is every $f\colon\mathR\to\mathR$ with (JC)
property
the product of two $\Ext$ functions?}

{\bf Open.}


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