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\title{Quasicontinuous functions with a little symmetry are extendable}



\author{{\small Francis Jordan}%
\thanks{AMS classification numbers: Primary 26A15;  Secondary 54C30
\endgraf  Key words and phrases: symmetrically continuous
functions,  extendable functions, quasicontinuous functions, peripherally
continuous functions, Darboux functions.
\endgraf},
\small  Department of Mathematics, University of Louisville,\\ 
Louisville, KY 40292}

\date{}


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%\title{The minimum number of Darboux functions needed to cover Baire
%class 1}
%\author{Francis Jordan\\
%\small Dept. of Mathematics, West Virginia Univ., Morgantown, WV
%26506-6310}

\date{}


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\begin{document}\maketitle

\begin{abstract}
It is shown that if a function $f\colon\real\to\real$ is quasicontinuous
and has a graph which is bilaterally dense in itself,
then
$f$ must be extendable to a connectivity function $F\colon\real^2\to\real$
and the set of discontinuity points of $f$ is $f$-negligible.  This
improves a result of H.~Rosen.  A similar result for symmetrically
continuous functions follows immediately. 
\end{abstract}
\section{Introduction}
In \cite{Rosen1} H.~Rosen proves the following theorem:
\thm{thm:ros}{If $f$ is Darboux, quasicontinuous and has a graph 
whose closure is bilaterally dense in itself, then $f$ is extendable and 
$\dcont(f)$ is $f$-negligible.}  
Our purpose here is to show that
the assumption in the theorem above that
$f$ be Darboux is redundant.  A corollary of this fact this that
symmetrically continuous quasicontinuous functions satisfy the conclusion 
of Theorem~\ref{thm:ros}.  
\section{Terminology}
For a set $S\subseteq\real$ we denote its closure by $\cl(S)$.     
Given $f\in\real^{\real}$ and $x\in\real$ we define
the right cluster set of 
$f$ at $x$ by 
\begin{equation*}
\wacc^{+}(f,x)=\bigcap_{n=1}^{\infty}(\cl(f[(x,x+1/n)])).   
\end{equation*} We define the left cluster set of $f$ 
at $x$, denoted   by $\wacc^{-}(f,x)$, in a similar way.  Finally,
we let 
$\wacc(f,x)=\wacc^{-}(f,x)\cap\wacc^{+}(f,x)$ denote  the
bilateral cluster set of $f$ at $x$.   

We say
$f\colon\real\to\real$ is a {\em cliquish} function provided that for every
$x_0\in\real$ and $\epsilon>0$ and neighborhood 
$W$ of $x_0$ there is a nonempty open set $W_0\subseteq W$ such that 
$\osc(f,{W_0})<\epsilon$.   We
denote the family of cliquish functions by $\cliq$.  It is well known, 
and easy to prove, 
that if $f\in\cliq$ then $\cont(f)$ is a co-meager subset of $\real$.  

We say a function $f$ has {\em closure bilaterally dense in itself} provided
that for every $x\in\real$ we have $\wacc^{+}(f,x)=\wacc^{-}(f,x)$.  
We will also be concerned with the following families of
functions.  We give descriptions of these families for 
general topological spaces, although our discussion will be  restricted
to the real line.  To find out more about the families below see
\cite{GNsurvey}, \cite{NATsurvey}, and 
\cite{KCsurvey}.
\begin{description}
\item[$\dar$:] $f\in Y^X$ is a {\em Darboux} function if and only if
$f[C]$ is connected  in $Y$ for every connected subset $C$ of $X$.
\item[$\conn$:] $f\in Y^X$ is a {\em connectivity} function if and only
if the graph of 
$f$ restricted to $C$ is connected in $X\times Y$ for every connected 
subset $C$ of $X$.
\item[$\ext$:] $f\in Y^X$ is an {\em extendable} function if and only if 
there is a connectivity function $g:X\times [0,1]\to Y$ such that 
$f(x)=g(0,x)$ for every $x\in X$.
\item[$\phc$:] $f\in Y^X$ is a {\em peripherally continuous} function 
if and only if  for every $x\in X$ and every pair of open sets $U\subset X$ and 
$V\subset Y$  such that $x\in U$ and $f(x)\in V$ there is an  open
neighborhood $W$ of $x$ with $\cl(W)\subset U$ and 
$f[\bd(W)]\subseteq V$, where $\bd(W)$ denotes the boundary of $W$.
\item[$\quasi$:] $f\in Y^X$ is a {\em quasi-continuous} function  if and
only if  at each point $p\in X$ the following condition holds:  for every
open set $U\subseteq X$ with 
$p\in U$ and open set $V\subseteq Y$ with $f(p)\in V$ there exists  a
non-empty open set $W\subseteq U$ such that $f[W]\subseteq V$.
\end{description}

It is clear from the definitions that $\quasi\subseteq\cliq$.  Given an
extendable function $f\colon\real\to\real$ we say that $A\subseteq\real$ is 
{\em $f$-negligible} if for any function $g\colon\real\to\real$ such that 
$g=f$ on
$\real\setminus A$ and $g|_{A}\subseteq\cl(f)$ we have that $g$ is also
extendable.
 
\section{The Results}
\thm{thm:1}{If $f$ is quasi-continuous and has a graph 
whose closure is bilaterally dense in itself, then $f$ is extendable and 
$\dcont(f)$ is $f$-negligible.}
\proof
Notice that quasi-continuity together with 
the property that the closure of
$f$ is bilaterally dense in itself implies that $f$ is peripherally
continuous.  

By Theorem~\ref{thm:ros} it is enough for us to show that $f$ is Darboux.  
Let $[a,b]$ be an interval such that
$f(a)\neq f(b)$ and $c$ be strictly between $f(a)$ and $f(b)$. 
Let $A=\{x\in [a,b]\colon f(x)>c\}$ and 
$B=\{x\in [a,b]\colon f(x)<c\}$.  By way of contradiction, assume that
$c\notin f[(a,b)]$.

Let $P=\cl(A)\cap\cl(B)$.  Suppose there is an isolated
point $x\in P$.  First notice that $x$ cannot be an endpoint of $[a,b]$.  
For example, if $x=a$ and $x\in B$, then the only way $x$ could be
isolated in $P$ would be if there were an $\epsilon>0$ such
that $[a,a+\epsilon)\subseteq A$, but this is impossible by peripheral
continuity.  Thus, $x$ is not an endpoint of $[a,b]$.  Since $x$ is
isolated in $P$, there is a $\delta>0$ such that 
($(x-\delta,x)\subseteq A$ or $(x-\delta,x)\subseteq B$) and
($(x,x+\delta)\subseteq A$ or $(x,x+\delta)\subseteq B$).  Without loss of
generality assume that $(x-\delta,x)\subseteq A$.  By peripheral continuity,
$x\notin B$.  Since $x\in\cl(B)$, it follows that $(x,x+\delta)\subseteq
B$.  Thus, $\wacc^{-}(f,x)\subseteq [c,+\infty)$ and
$\wacc^{+}(f,x)\subseteq (-\infty,c]$.  Since $f$ has closure bilaterally
dense in itself, it follows that $\wacc(f,x)=\{c\}$.  In which case, by
peripheral continuity, we would have $f(x)=c$.  Thus, we may assume that $P$
has no isolated points.  

Suppose that $P$ is not nowhere dense.  By cliquishness, 
if $P$ contains a nontrivial interval, then
$P$ must contain a continuity point $x$ of $f$.  It is easy
to see that $f(x)=c$.  Thus, we may assume that $P$ is nowhere dense. 

The set $\cl(A)\cap\cl(B)$ is nonempty, otherwise
$[a,b]\setminus (\cl(A)\cup\cl(B))$ would contain a nonempty open interval
upon which $f$ would be constantly equal to $c$, contrary to our
assumption.   

So, we may assume $P$ is a nonempty perfect set.  
Notice that by quasi-continuity, the sets $A\cap I$ and $B\cap I$ are
not contained in $P\cap I$ for any nontrivial open interval $I\subseteq
[a,b]$ such that $I\cap P\neq\emptyset$.  Let
$x_0\in P$ be the endpoint of some interval contained in $A$.  Since the
closure of
$f$ is bilaterally dense in itself and $f[A]\subseteq (c,+\infty)$ there is a
$\delta_0>0$ such that $f[(x_0-\delta_0,x_0+\delta_0)]\subseteq
(c-1,+\infty)$.   Since $P$ is perfect, there is an $x_1\in
(x_0-\delta_0,x_0+\delta_0)$ which is the endpoint of an open interval
contained in $B$.  Since the closure of
$f$ is bilaterally dense in itself and $f[B]\subseteq (-\infty,c)$, there is
a
$\delta_1>0$ such that $f[(x_1-\delta_1,x_1+\delta_1)]\subseteq
(-\infty,c+1/2)$ and $[x_1-\delta_1,x_1+\delta_1]\subseteq
(x_0-\delta_0,x_0+\delta_0)$.  Continue inductively defining $\delta_n$ and
$x_n$ so that the following conditions hold:
\begin{description}
\item[(1)] $x_n$ is an endpoint of an interval contained in $A$ ($B$) if $n$
is even (odd),
\item[(2)] $[x_n-\delta_n,x_n+\delta_n]\subseteq
(x_{n-1}-\delta_{n-1},x_{n-1}+\delta_{n-1})$,
\item[(3)] if $n$ is even, then $f[(x_n-\delta_n,x_n+\delta_n)]\subseteq
(c-1/2^{n},+\infty)$,
\item[(4)] if $n$ is odd, then $f[(x_n-\delta_n,x_n+\delta_n)]\subseteq
(-\infty,c+1/2^{n})$ and 
\item[(5)] $\delta_n<1/2^{n}$. 
\end{description}
Since $P$ is closed and $\lim_{n\to\infty}\delta_n=0$, there is a point $x\in
P$ such that $x\subseteq [x_n-\delta_n,x_n+\delta_n]$ for every
$n\in\omega$.   By (3) and (4) we must have $f(x)=c$.  Thus,
$f$ is a Darboux function completing the proof.\qed

Since a symmetrically continuous function must have a graph with closure
bilaterally dense in itself, we have the following corollary of
Theorem~\ref{thm:1}:
\cor{cor:1}{If $f$ is quasicontinuous and symmetrically continuous, then  
$f$ is extendable and $\dcont(f)$ is $f$-negligible.}

I do not know if peripheral continuity alone is sufficient to guarantee that
a symmetrically continuous function is extendable or even Darboux.


\begin{thebibliography}{22}


\bibitem{KCsurvey} K.~Ciesielski, Set Theoretic Real Analysis, {\it J.
Appl. Anal.}, {\bf 3}({\bf 2}) (1997), 143--190. (Preprint* available.)
\footnote{Preprints marked by * are available in electronic
form.  They can be accessed from {\em Set Theoretic Analysis Web Page}:
http://www.math.wvu.edu/homepages/kcies/STA/STA.html.}  


\bibitem{GNsurvey} R. G.~Gibson and T.~Natkaniec, Darboux-like Functions, 
{\it Real Anal.  Exchange} {\bf 22}({\bf 2}) (1996-97), 492--533.


\bibitem{NATsurvey} T.~Natkaniec, Almost Continuity,  {\it Real Anal.
Exchange} {\bf 17}(1991-92), 462--520.

\bibitem{Rosen1} H.~Rosen, Darboux Quasicontinuous Functions,  {\it Real
Anal. Exchange} {\bf 23}(1997-98), 631--640.

\end{thebibliography}

\end{document}