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\documentclass{rae}

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

%\coverauthor{Richard G. Gibson and Tomasz Natkaniec}
%\covertitle{Darboux like functions}

\received{March 3, 1997}

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

\keywords{Darboux functions, extendable functions, almost
continuous functions, connectivity functions, functions with
 perfect road, peripherally continuous functions, DIVP-functions,
CIVP-functions, SCIVP-functions, WCIVP-functions,
Sierpi{\'n}ski-Zygmund functions, property~($B$)}

\firstpagenumber{491}

\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, 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}


%%%%%%Put Author's Definitions Below Here %%%%%%%%%%%

\newtheorem{Th}{Theorem}[section]
\newtheorem{Le}[Th]{Lemma}
\newtheorem{Co}[Th]{Corollary}
\newtheorem{Po}[Th]{Proposition}
\newtheorem{Ro}[Th]{Remark}
\newtheorem{Qu}[Th]{Question}
\newtheorem{Pro}[Th]{Problem}
\newcommand\mathN{{\mathbb N}}
\newcommand\mathI{{\mathbb I}}
\newcommand\mathR{{\mathbb R}}
\newcommand\mathQ{{\mathbb Q}}
\newcommand\real{\mathR}
\newcommand{\RRR}{\mathR^{\mathR}}
\newcommand{\restr}{{\restriction}}
\newcommand{\co}{{\mathfrak c}}
% \newcommand{\Qed}{\unskip\nolinebreak\quad\hfill$\Box\;\;$\medskip}
\def\Qed{\qed}

\def\PO{{\cal P}(\mathR)}
\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}}
\newcommand{\SZ}{{\rm SZ}}

\newcommand\cf{{\rm cf}}
\newcommand\id{{\rm id}}
\newcommand\cl{{\rm cl}\,}
\newcommand\Cl{{\rm cl}\,}
\newcommand\rng{{\rm rng}\,}
\newcommand\dom{{\rm dom}\,}
\newcommand\bd{{\rm bd}\,}
\newcommand\Int{{\rm int}\,}
\newcommand{\pf}{{\noindent\sc Proof. }}
\newcommand{\charf}[1]{\mbox{\raise.48ex\hbox{$\chi$}$_{#1}$}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\maketitle
\tableofcontents

\section{Historical background}
 A function $f:\mathR\to\mathR$ is said to have the {\em
 intermediate value property} provided that if $p$ and $q$ are
 real numbers such that $p\neq q$ and $f(p)< f(q)$, then for
 every $y\in (f(p),f(q))$ there exists a number $x$ between $p$
 and $q$ with $f(x)=y$. In 1875, G. Darboux showed that there
 exist functions with the intermediate value property that are
 not continuous~\cite{22}.  Because of his work with functions
 having the intermediate value property, these functions are
 called Darboux functions.

In 1907, J. Young~\cite{82} studied real-valued functions defined
 on an interval with the following property: for every
 $x\in\mathR$ there exist sequences $\{x_n\}$ and $\{y_n\}$ such
 that $x_n\nearrow x$, $y_n\searrow x$, and both $ f(x_n)$ and
 $f(y_n)$ converge to $f(x)$. In~\cite{82}, J. Young showed that
 for Baire class~1 functions, Darboux functions and functions
 having this property of Young are equivalent. In more general
 spaces, functions having the property of Young are said to be
 peripherally continuous. (See \cite{39,40,81}.)

K.~Kuratowski and W.~Sierpi{\'n}ski, in 1922, showed that for real-valued
Baire class~1 functions defined on an interval, Darboux functions and
functions with a connected graph are equivalent~\cite{54}.

I.~Maximoff, in 1936, showed that for real-valued Baire class~1
 functions defined on an interval, Darboux functions and
 functions with a perfect road are equivalent~\cite{59}.

J.~Stallings, in 1959, defined almost continuous functions in the
 sense of Stallings and extendable functions~\cite{77}.

 In~\cite{9}, J.~Brown showed that for real-valued Baire class~1
 functions defined on an interval, Darboux functions and almost
 continuous functions in the sense of Stallings are equivalent.
 In~\cite{11}, J.~Brown, P.~Humke, and M.~Laczkovich showed that
 for real-valued Baire class~1 functions defined on an interval,
 Darboux functions and extendable functions are equivalent.

From the preceding we see that for Baire class~1 functions
 $f:\mathR\to\mathR$, Darboux functions have been characterized
 in several ways. For further information concerning these
 functions see the first three chapters of the book~\cite{12} by
 A.~M.~Bruckner.


\section{Basic definitions}

Our terminology is standard. We consider only real-valued
 functions of one real variable. No distinction is made between
 a function and its graph. By $\mathR$ and $\mathI$ we denote the set
of all reals and the interval $[0,1]$, respectively.
 The family of all subsets of a set $X$ is denoted by $\P(X)$.
 The family of all functions from a set $X$ into $Y$ is denoted
 by $Y^X$.  By $\CC$ and Const we denote the families of all
 continuous functions and all constant functions. The symbol
 $|X|$ stands for the cardinality of a set $X$. The cardinality
 of $\mathR$ is denoted by $\co$. For the cardinal number
 $\kappa$ we write $[X]^{\kappa}$ to denote the family of all
 subsets $Y$ of $X$ with $|Y|=\kappa$. In particular, $[X]^1$
 stands for the family of all singletons in $X$ and $[X]^2$ for
 the family of all doubletons in $X$.  By a Cantor set we mean
 any non-empty perfect nowhere dense subset of $\mathR$.
 Moreover, we say that a set $A\subset\mathR$ is Cantor dense in
 a set $X\subset\mathR$, if $A\cap J$ contains a Cantor set
 whenever $J$ is a non-empty open interval $J$ with $J\cap
 X\neq\emptyset$. By $(a,b)$ we denote an open interval with
 end-points $a$ and $b$, i.e., the set of all $x\in\mathR$ such
 that $\min\{a,b\}<x<\max\{a,b\}$.


The following is a list of the definitions of the different types
 of functions that will be investigated. Note that we have
 abbreviated these classes of functions with letters on the left.

Let $X$ and $Y$ be topological spaces and let $f:X\to Y$ be a
function. Then:

\begin{description}
\item[D]
 -- $f$ is a {\it Darboux function} if $f(C)$ is connected
 whenever $C$ is connected in $X$;
\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 $W$;
\item[Conn]
 -- $f$ is a {\it connectivity function} if the graph of $f$
 restricted to $C$, denoted by $f\restr C$, 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 $g:X\to Y$~\cite{77};
\item[Ext]
 -- $f$ is an {\it extendable function} if there exists a
 connectivity function $g:X\times
\mathI\to Y$ such that $f(x)=g(x,0)$ for all $x\in X$~\cite{77}.
\end{description}
%ch
The class of all Darboux functions from $X$ to $Y$ we shall denote by
$\D(X,Y)$, or shortly, by $\D$, when $X$ and $Y$ will be clear from a
context (usually, $X=Y=\mathR$). Similarly, for other Darboux like classes.


The next definitions concern only real-valued functions defined
 on $\mathR$ (or, on subspaces of $\mathR$).  Then:
\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 $x$~\cite{59};
\item[WCIVP]
 -- {\it Weak Cantor Intermediate Value Property}: $f\in\WCIVP$
 if for all $p,q\in \mathR$ with $p<q$ and $f(p)\neq f(q)$, there
 exists a Cantor set $C\subset (p,q)$ such that $f(C)$ is between
 $f(p)$ and $f(q)$~\cite{30};
\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 $f(C)\subset
 K$~\cite{29};
\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~\cite{68};
\item[PB]
 -- {\it Property B}: $f\in \PB$ if for all pairs of open
 intervals $I$ and $J$, if $I\cap f^{-1}(J)$ is uncountable, then
 $I\cap f^{-1}(J)$ contains a non-empty perfect set~\cite{PB}.
\end{description}

\begin{Th}\label{t1}
If $f:\mathR\to\mathR$, then
\begin{itemize}
\item
$f\in\D$ if and only if $f$ has the intermediate value property;
\item
 $f$ is a connectivity function if and only if the entire graph
 of $f$ is connected;
\item
 $f$ is peripherally continuous if and only if it satisfies the
 Young condition:
\begin{quote}
 for every $x\in\mathR$ there exist sequences $\{x_n\}$ and
 $\{y_n\}$ such that $x_n\nearrow x$, $y_n\searrow x$, and both
 $f(x_n)$ and $f(y_n)$ converge to $f(x)$.
\end{quote}
\item
 $\Conn\subset\D\subset\PC\,\,\,\mbox{ and }\,\,\,
\PR\subset\PC$
\end{itemize}
\end{Th}

The preceding is a brief survey of some of the known facts
 concerning the families of functions defined.

\section{Darboux like functions in the class $\RRR$}
 Stallings defined almost continuous functions in the sense of
 Stallings ($\ACS$) in~\cite{77} and proved the implications
 $$\Ext\subset\ACS\;\;\;\mbox{ and }\;\;\;\ACS\subset\Conn.$$
 Thus in the class of all functions $\RRR$, we have the following
 implications.

 \begin{picture}(0,50)
 \put(20,35){\makebox(0,0){$\CC$}}
 \put(35,35){\vector(1,0){20}}
  \put(75,35){\makebox(0,0){$\Ext$}}
 \put(88,35){\vector(1,0){20}}
 \put(120,35){\makebox(0,0){${\ACS}$}}
 \put(135,35){\vector(1,0){25}}
 \put(180,35){\makebox(0,0){$\Conn$}}
 \put(195,35){\vector(1,0){20}}
 \put(224,35){\makebox(0,0){${\D}$}}
 \put(233,35){\vector(1,0){35}}
 \put(280,35){\makebox(0,0){${\PC}$}}
   \put(180,10){\makebox(0,0){${\PR}$}}
\put(190,10){\vector(4,1){80}}
 \put(90,31){\vector(4,-1){80}}
\end{picture}
\begin{center} Chart~1 \end{center}\label{chartone}

Note that all inclusions in Chart~1, denoted by $\to$, are
 proper. (See~\cite{11}.)

\medskip
The following contains a survey of research that has been done
 concerning these properties. In this investigation a series of
 questions were asked by the first of the Authors in a lecture
 given at the Banach Center in November of 1989~\cite{26}. We
 will now give an update on the status of those questions. The
 first question:
\begin{Qu}\label{q1}
 What is both a necessary and sufficient condition for an almost
continuous function in the sense of Stallings to be an extendable function?
\end{Qu}
 \noindent is one on which the first Author has spent a great amount of
 time. This question also led to most of the others questions.

\medskip
F. Roush and R. Gibson in their investigation of the
 characterization of extendable functions defined the properties
 $\WCIVP$, $\CIVP$, and $\SCIVP$.
 (See~\cite{27,29, 30,68}.) These properties were given the
 following names due to their similarity to the intermediate
 value property. Hence they are Darboux like functions.

In reference to the question
\begin{center}
``Does $\ACS\subset\Ext$?''
\end{center}
 asked by Stallings in~\cite{77}, Gibson and Roush in~\cite{30},
 defined the weak Cantor intermediate value property($\WCIVP$),
 and proved the following results.
\begin{Th}{\rm (Gibson, Roush~\cite{30})}\label{t3}
In the class $\mathI^{\mathI}$ we have
\begin{itemize}
\item
if $f\in\Ext$, then $f\in\WCIVP$,
 \item
 there exists a function $f:{\mathI}\to {\mathI}$ such that
 $f\in\ACS$ and $f\notin\WCIVP$.
\end{itemize}
\end{Th}

Hence in the class of all real functions defined on subintervals
 of $\mathR$ we have
\begin{Co}\label{c3}
$\Ext\subset (\ACS\cap\WCIVP)\;\;\mbox{ and }
\ACS\neq\Ext.$
\end{Co}

\begin{Th}{\rm (Gibson, Roush~\cite{33})}\label{t4}
In the class ${\mathI}^{\mathI}$ we have
\begin{itemize}
\item
if $f\in\Ext$ then $f\in\PR$;
\item
there exists a function $f:{\mathI}\to {\mathI}$
such that $f\in\ACS$ but $f\notin\PR$.
\end{itemize}
\end{Th}

Hence in the class of all real functions defined on subintervals
 of $\mathR$ we have
\begin{Co}\label{c1}
$\Ext\subset\ACS\cap\PR.$
\end{Co}

In \cite{33}, Gibson and Roush posed the following question:
\begin{quote}
 ``Does there exists a function $f\colon {\mathI}\to {\mathI}$
 such that $f\in \ACS\cap\PR$ but $f\notin \Ext$?''.
\end{quote}

Rosen, Gibson, and Roush in~\cite{68} gave an affirmative answer
 to this question by proving the following theorem

\begin{Th}{\rm (Rosen, Gibson, Roush~\cite{68}) \label{c5}}
In the class ${\mathI}^{\mathI}$ we have
\begin{itemize}
\item
 if $f\in\Ext$, then $f\in\SCIVP$;
 \item
 there is a function $f\in (\ACS\cap\PR)\setminus\CIVP$.
 \end{itemize}
 \end{Th}
 Since $\SCIVP\subset\CIVP$, hence in the class of all real
 functions defined on subintervals of $\mathR$ we have
\begin{Co}\label{c2}
 $\Ext\neq (\ACS\cap\PR).$
 \end{Co}

It was stated in~\cite{33} that $\CIVP\subset\PR$ but it was not
 proved. The proof of that statement now follows.

\begin{Th}\label{t6}
 If $f\colon\mathR\to\mathR$ is a function and $f\in\CIVP$, then $f\in\PR$.
\end{Th}
\pf
 Select any $x\in\mathR$. Assume that there exists
 $\varepsilon>0$ such that $f$ is constant on no subinterval of
 $[x-\varepsilon ,x]$ having $x$ as a right endpoint.

Let $x_n$ be a increasing sequence in $(x-\varepsilon ,x)$ such
 that $x_n\to x$, $f(x_n)\neq f(x)$ and $f(x_n)\to f(x)$.
 Select any Cantor set $K_n$ between $f(x)$ and $f(x_n)$ such
 that $K_n$ is a subset of $(f(x)-1/n,f(x)+1/n)$. Since
 $f\in\CIVP$, there exists a Cantor set $C_n$ between $x_n$ and
 $x$ such that $f(C_n)$ is a subset of $K_n$. Let $A$ be the
 union of all $C_n$ and $\{ x\}$.  Then $A$ is a
 perfect set and $f\restr A$ is continuous at $x$ (from the
 left). In a similar way we can construct a perfect set $B$ such
 that $f\restr B$ is continuous at $x$ (from the right).

If $f$ is constant on $[x-\varepsilon ,x]$ or $[x,
x+\varepsilon ]$ for some $\varepsilon>0$, let $A=[x-\varepsilon
, x ]$ or $B=[x,x+\varepsilon ]$. Now if $P=A\cup
 B$, then $P$ is a perfect set with $x$ as a bilateral limit
 point and $f\restr P$ is continuous at $x$.
\Qed

For real-valued functions $f:R\rightarrow R$ we have only the
 following implications among the classes of functions defined
 above. This is an expansion of the previous diagram. (See the
 papers by Brown, Humke and Laczkovich,~\cite{11}; Rosen, Gibson
 and Roush~\cite{68}; and Banaszewski and Natkaniec,~\cite{3}.)

   \begin{picture}(0,100)
 \put(20,55){\makebox(0,0){$\CC$}}
 \put(30,55){\vector(1,0){20}}
  \put(65,55){\makebox(0,0){$\Ext$}}
  \put(120,55){\makebox(0,0){$\ACS\cap\PR$}}
    \put(190,55){\makebox(0,0){$\WCIVP$}}
    \put(78,55){\vector(1,0){15}}
\put(150,55){\vector(1,0){15}}
\put(150,53){\vector(3,-1){55}}
 \put(115,60){\vector(0,1){12}}
 \put(120,80){\makebox(0,0){${\ACS}$}}
 \put(135,80){\vector(1,0){20}}
 \put(170,80){\makebox(0,0){$\Conn$}}
 \put(185,80){\vector(1,0){20}}
 \put(214,80){\makebox(0,0){${\D}$}}
 \put(223,78){\vector(1,-1){20}}
 \put(260,55){\makebox(0,0){${\PC}$}}
 \put(117,30){\makebox(0,0){${\SCIVP}$}}
\put(175,35){\vector(1,1){15}}
  \put(170,30){\makebox(0,0){${\CIVP}$}}
   \put(215,30){\makebox(0,0){${\PR}$}}
 \put(225,28){\vector(1,-1){20}}
\put(135,30){\vector(1,0){20}}
 \put(185,30){\vector(1,0){20}}
 \put(78,52){\vector(1,-1){20}}
 \put(223,35){\vector(1,1){20}}
   \put(260,5){\makebox(0,0){${\PB}$}}
\end{picture}
\begin{center} Chart~2 \end{center}

Note that in the class $\RRR$ all inclusions in Chart~2 are proper.

We now recall some additional questions from~\cite{26} and give
 the answer, if it is known.

\begin{Qu}
If $f\in\CIVP$, is $f\in\SCIVP$?
\end{Qu}
Answer: {\bf No.}
K.~Banaszewski and T.~Natkaniec in~\cite{3} constructed a
 Sierpi{\'n}ski-Zygmund ($\SZ$) function $f\colon\mathR\to\mathR$
 having the $\CIVP$ and observed that every function in $\SZ$
 does not have the $\SCIVP$. (See Theorem~\ref{BN}.) Thus
 $f\in\CIVP\setminus\SCIVP$.

\begin{Qu}
If $f\in (\ACS\cap\CIVP)$, is $f\in\Ext$?
\end{Qu}
%ch
 Answer: {\bf No.} If the real line $\mathR$ is not a union of
 less than continuum many of its meager subsets, K.~Banaszewski and
T.~Natkaniec in
 \cite{3} constructed $f\in (\ACS\cap\CIVP\cap\SZ)$. (See
 Theorem~\ref{BN1}.) Since $f\in\SZ$, $f\notin
\SCIVP$. Therefore $f\notin\Ext$. Moreover, quite recently,
K.~Ciesielski~\cite{kc-nowe} constructed in ZFC an example $f\in
(\ACS\cap\CIVP)$ that is continuous on no perfect subset. Thus
$f\not\in\SCIVP$ and consequently, $f\not\in\Ext$.

The next question remains {\bf open}\footnote{Recently H.~Rosen proved
under CH that there exists $f\in\ACS\cap\SCIVP\setminus\Ext$~\cite{HRn}.
Actually, his proof works under assumption that the union of less than
$\co$ many meger sets is meger.}.
\begin{Qu}
If $f\in (\ACS\cap\SCIVP)$, is $f\in\Ext$?
\end{Qu}

In \cite{27} it was shown that if $f:[a,b]\to\mathR$ and $f\in \D$, then
$$\WCIVP=\PB=\PR.$$
 Also we discussed the following questions which have negative
answers.

\begin{Qu}
If $f\in (\ACS\cap\PR)$, is $f\in\Ext$?
\end{Qu}
Answer: {\bf No.} See Corollary~\ref{c2}.

\begin{Qu}\label{q6}
 If $f\in (\Conn\cap\PR)$, is $f\in\ACS$?
 \end{Qu}
 Answer:  {\bf No.} See \cite{11} and \cite{27}.

\begin{Qu}\label{q7}
 If $f\in\D\cap\PR$, is $f\in\Conn$?
 \end{Qu}
  Answer: {\bf No.} See \cite{11} and \cite{27}.

Each of the functions defined in the answers to
 Questions~\ref{q6} and \ref{q7} has a graph that is a $G_\delta$
 set, and hence is Borel measurable. Thus they satisfy the
 $\SCIVP$. (See subsection~\ref{borel}.)

However {\bf we left open} the following question.
\begin{Qu}\label{Gd}
 If $f\in\ACS$ and has a $G_\delta$ graph, is $f\in\Ext$?
\end{Qu}

\subsection{Darboux like functions in the first class of Baire}
 Darboux like functions that belong to the first class of Baire
 were studied in many papers. (See also the survey of J.~Ceder and
 T.~L.~Pearson~\cite{CP}.)

\begin{Th}{\rm (See~\cite{12}.)}\label{t11}
In Baire class one the following properties are equivalent:
$$\Conn=\D=\PR=\PC.$$
\end{Th}

\begin{Th}{\rm (Brown~\cite{9})}\label{t12}
In Baire class one,
$$\ACS=\Conn.$$
\end{Th}

\begin{Th}{\rm (Brown, Humke, Laczkovich~\cite{11})}
\label{t13}
In Baire class one,
$$\Ext=\ACS.$$
\end{Th}

\begin{Co}\label{c11}
In Baire class one the following properties are equivalent
$$\Ext=\ACS=\Conn=\D=\SCIVP= \CIVP=\PR=\PC.$$
\end{Co}

Now we will state the relations that hold in the first class of
 Baire between Darboux functions and $\PB$ and $\WCIVP$
 functions.

First, because every Borel measurable function has the property $\PB$, so
$$\PC\cup\WCIVP\subset\PB.$$
 On the other hand, the characteristic function of the halfline
 $(0,\infty)$ belongs to $\PB\setminus(\PC\cup\WCIVP)$. Thus
$$\PB\not\subset\PC\;\;\;\mbox{ and }\;\;\;\PB\not\subset\WCIVP.$$

\subsection{Darboux like functions that are Borel measurable}\label{borel}
 For Borel measurable function, Brown, Humke and Laczkovich
 proved the following theorem.

\begin{Th}{\rm (Brown, Humke, Laczkovich~\cite{11})}
 In the class of Borel measurable functions the following
 implications hold
 $$\Ext\Rightarrow\ACS\Rightarrow\Conn\Rightarrow
\D\Rightarrow\PR\Rightarrow\PC.$$
 Moreover, those implications are not reversible except for
 possibly $\Ext\Rightarrow\ACS$.
\end{Th}

Thus we have the following {\bf open question}. (See also
 Question~\ref{Gd}.)
\begin{Qu}\label{Bor}
 If $f:{\mathI}\to {\mathI}$ is a Borel measurable function and
 $f\in\ACS$, is $f\in\Ext$?
\end{Qu}

The next example is strictly connected with the Question~\ref{Bor}.

\noindent
{\bf Example. }(Ces{\'a}ro)\label{Vietoris}
Let $\varphi:{\mathI} \to {\mathI}$ be defined by
$$\varphi(x)= \overline{\lim}_{n
\to\infty}\frac{a_{1}+ \ldots +a_{n}}{n}$$
 where $a_{i}$ are given by the unique nonterminating binary
 expansion of the number $x=(0.a_{1}a_{2} \ldots )$.


The function $\varphi$ is called the {\it Ces{\'a}ro-Vietoris}
 function. Note that $\varphi$ belongs to the second class of
 Baire~\cite{12}. Vietoris proved in 1921 that $\varphi$ is
 connected: $\varphi\in\Conn$~\cite{78}. In 1975, J.~Brown proved
 that $\varphi\in\ACS$~\cite{10}. The following problem remains
 {\bf open}:

\begin{Qu}\label{CV}
Does the Ces{\'a}ro-Vietoris function $\varphi$ belong to $\Ext$?
\end{Qu}

Note that the solution of Question~\ref{CV} in the negative
 implies also the negative answer to the Question~\ref{Bor}.

Note also that in the class of Borel measurable functions,
$$\D\subset\SCIVP=\CIVP.$$
Thus, $\ACS\cap\PR=\ACS$.

\subsection{Darboux like functions that are Lebesgue measurable}

\begin{Th}{\rm (Brown, Humke, Laczkovich~\cite{11})}
 In the class of all Lebesgue measurable functions the following
 relations hold:

   \begin{picture}(0,50)
 \put(20,35){\makebox(0,0){$\CC$}}
 \put(35,35){\vector(1,0){20}}
  \put(75,35){\makebox(0,0){$\Ext$}}
 \put(88,35){\vector(1,0){20}}
 \put(120,35){\makebox(0,0){${\ACS}$}}
 \put(135,35){\vector(1,0){25}}
 \put(180,35){\makebox(0,0){$\Conn$}}
 \put(195,35){\vector(1,0){20}}
 \put(224,35){\makebox(0,0){${\D}$}}
 \put(233,35){\vector(1,0){35}}
 \put(280,35){\makebox(0,0){${\PC}$}}
   \put(180,10){\makebox(0,0){${\PR}$}}
\put(190,10){\vector(4,1){80}}
 \put(90,31){\vector(4,-1){80}}
\end{picture}

\noindent
Moreover, all those inclusions are proper.
\end{Th}

\subsection{Darboux like functions that are Marczewski measurable}

Recall that a function $f\colon X\to Y$ is said to have {\it
 property-(s)} or to be {\it (s)-measurable} provided that
\begin{description}
\item[(s)]
-- for each
 non-void perfect subset $P$ of $X$ there exists a non-void
 perfect subset $Q$ of $P$ such that the restriction $f\restr Q$
 is continuous.
\end{description}
 Marczewski defined property $(s)$ for sets in~\cite{58} and showed that
the class of
 $(s)$-measurable ({\it Marczewski measurable}) functions and the class of
 functions (functions with property $(s)$) studied by
 Sierpi{\'n}ski in~\cite{73} were the same. Note that each Borel
 measurable function is Marczewski measurable.

\begin{Th}{\rm (Gibson, Roush~\cite{34})}
There exists a connectivity function $g\colon {\mathI}^2\to {\mathI}$
 and $p\in\mathI$ such that the extendable function
 $f\colon\mathI\to\mathI$ given by $f(x)=g(x,p)$
does not have property $(s)$.
Thus
 $f$ is not Marczewski measurable.
\end{Th}

J. B. Brown, P. Humke, and M. Laczkovich in~\cite{11} stated the
 problem that can be formulated as follows:
\begin{Qu}
 How are the Darboux like properties related within the function classes:
\begin{description}
\item[U]
-- universally measurable functions;
\item[B$_w$]
--  functions with the Baire property in wide sense;
\item[B$_r$]
-- functions with the Baire property in restricted sense;
\item[(s)]
-- Marczewski measurable functions?
\end{description}
\end{Qu}

See~\cite{53} for definitions and discussion. Generally, this
 problem remains {\bf open}.  However, we know that the answer
 depends on some additional set theoretical assumptions. Namely,
 concerning this problem, I.~Rec{\l}aw and R.~G.~Gibson,
 \cite{37}, proved the following theorems.

\begin{Th}{\rm (Gibson, Rec{\l}aw~\cite{37})}
For functions $f\colon\mathR\to\mathR$, the following are equivalent:
\begin{enumerate}
\item[(i)]
${\rm U}\cap\D\subset\PR$;
\item[(ii)]
${\rm U}\cap\ACS\subset\PR$;
\item[(iii)]
there is no universally null set of size of the continuum on the real line.
\end{enumerate}
\end{Th}

\begin{Th}{\rm (Gibson, Rec{\l}aw~\cite{37})}
For functions $f\colon\mathR\to\mathR$, the following are equivalent:
\begin{enumerate}
\item[(i)]
${\rm B}_r\cap\D\subset\PR$;
\item[(ii)]
${\rm B}_r\cap\ACS\subset\PR$;
\item[(iii)]
 there is no always of the first category set of size of the
 continuum on the real line.
\end{enumerate}
\end{Th}

\section{Darboux like properties in the class of
 Sierpi{\'n}ski-\-Zyg\-mund functions}

The next theorems are connected with a theorem of Blumberg from 1922.
\begin{Th}\label{Blumb}
 {\rm (Blumberg \cite{4})}
 For every $f\colon\mathR\to\mathR$ there exists a dense subset
 $D$ of $\mathR$ such that the restriction
$f\restr D$ of $f$ to $D$ is continuous.
\end{Th}

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.
 This theorem shows that we cannot prove
 in ZFC a version of the Blumberg theorem in which the set $D$ is
 uncountable.

\begin{Th}\label{sz}
 {\rm (Sierpi\'nski, Zygmund \cite{74})}
There exists a function $f\colon\mathR\to\mathR$
whose restriction $f\restr X$ is discontinuous for any
subset $X$ of $\mathR$ of cardinality $\co$.
\end{Th}

Every function that satisfies the assertions of Theorem~\ref{sz}
 is called to be a {\it Sierpi{\'n}ski-Zygmund} function
 (shortly, $\SZ$-function):
\begin{description}
\item[SZ]
 -- $f$ is $\SZ$-function if the restriction $f\restr X$ is
 discontinuous for any subset $X$ of $\mathR$ of cardinality
 $\co$.
\end{description}

In 1981, J.~Ceder constructed an example of connectivity $\SZ$
 function.
\begin{Th}\label{Ceder}
{\rm (Ceder~\cite{17})} Assume the Continuum Hypothesis (CH). Then
$$\SZ\cap\Conn\neq\emptyset.$$
\end{Th}

This result was improved by K.~Kellum.
\begin{Th}\label{Kellum}
{\rm (Kellum~\cite{51})} Assume the Continuum Hypothesis (CH). Then
$$\SZ\cap\ACS\neq\emptyset.$$
\end{Th}

On the other hand, it is easy to observe that
$$\SZ\cap\SCIVP=\emptyset.$$
Thus
\begin{Th}
$\SZ\cap\Ext=\emptyset.$
\end{Th}

The $\PR$ functions in the class $\SZ$
were considered by Darji in 1993.
\begin{Th}\label{Darji}
{\rm (Darji~\cite{23})} There exists $f\in\SZ\cap\PR$.
\end{Th}

Answering a question posed by Darji, in 1996 Balcerzak,
 Ciesielski and Natkaniec proved the following theorem

\begin{Th}\label{BCN}
{\rm (Balcerzak, Ciesielski, Natkaniec \cite{1})}
\begin{enumerate}
\item[(a)]
If $\mathR$ is not a union of less
 than continuum many of its meager subsets (thus under CH and MA)
 then there exists an $f\in\SZ\cap\PR\cap\ACS$.
\item[(b)]
 There is a model of ZFC in which every Darboux function
 $f\colon\mathR\to\mathR$ is continuous on some set of
 cardinality continuum.

In particular, in this model we have  $\SZ\cap\ACS=\SZ\cap\D=\emptyset$.
\end{enumerate}
\end{Th}

K.~Banaszewski and T.~Natkaniec replaced $\PR$ property in
 Theorem~\ref{Darji} by $\CIVP$.
\begin{Th}\label{BN}
{\rm (K.~Banaszewski, Natkaniec~\cite{3})}
There exists $f\in\SZ\cap\CIVP$.
\end{Th}

Thus we obtain the following.
\begin{Co}
$\SCIVP\neq\CIVP.$
\end{Co}

Similarly, part (a) of Theorem~\ref{BCN} is improved as follows.

\begin{Th}\label{BN1}
{\rm (K.~Banaszewski, Natkaniec~\cite{3})}
If $\mathR$ is not a union of less
 than continuum many of its meager subsets, then there exists an
 $f\in\SZ\cap\CIVP\cap\ACS$.
\end{Th}

\begin{Co}
If $\mathR$ is not a union of less
 than continuum many of its meager subsets, then
 $$\Ext\neq\ACS\cap\CIVP.$$
 \end{Co}

 \section{Darboux like and additive functions}
\newcommand{\Add}{{\rm Add}}
 In 1942, F.~B.~Jones constructed a function
 $f\colon\mathR\to\mathR$ such that
\begin{enumerate}
\item[(1)]
$f$ is additive, i.e., $f(x+y)=f(x)+f(y)$ for each $x,y\in\mathR$;
\item[(2)]
 $f$ intersects every closed subset $P$ of $\mathR^2$ with
 uncountable $x$-projection $\dom(P)$.
\end{enumerate}

Such a function was studied in several papers.
\begin{Th}
Let $f\colon\mathR\to\mathR$ be the Jones' function. Then
\begin{enumerate}
\item[(1)]
$f$ is connectivity; {\rm (Jones~\cite{46})}
\item[(2)]
$f$ is almost continuous in the sense of Stallings; {\rm (Kellum~\cite{51})}
 \item[(3)]
 $f$ does not have the $\WCIVP$, thus it is not extendable.
 {\rm (Rosen~\cite{69})}
 \end{enumerate}
 \end{Th}

Darboux like properties in the class {\bf Add} of additive
 functions were also considered by J.~Sm{\'\i}tal~\cite{JS} and
 by Z.~Grande~\cite{ZG}. Grande in his paper~\cite{ZG} posed the
 following, very interesting question. It was presented during
 the Joint US-Polish Workshop in Real Analysis in
 {\L}{\'o}d{\'z}, Poland, in July~1994, but still {\bf remains
 open}\footnote{Recently K.~Ciesielski and U.~Darji find under CH the
affirmative answer to this problem.(Private communication.)}. (See
also~\cite{GMN}.)
 \begin{Qu}
 Does there exist a discontinuous additive almost continuous in
 the sense of Stallings
 (or connected) function whose graph is ``small'' in the
 sense of measure or category?
\end{Qu}

Recall that there are discontinuous additive Darboux functions
 possessing small graph both in the sense of measure and in the
 sense of category.
(See~\cite{DB}.)

Recently Darboux like functions in the class $\Add$ were
 considered by D.~Banaszewski in his doctor's thesis. In
 particular, he proved the following
 \begin{Th} {\rm (D.~Banaszewski~\cite{DB})}
 For every $f\in\Add$ the following conditions are equivalent
 \begin{enumerate}
 \item[(i)]
 $f\in\PR$;
 \item[(ii)]
 $f$ has a perfect road at $0$;
 \item[(iii)]
 $f$ has a perfect road at some $x\in\mathR$;
 \item[(iv)]
 $f\in\WCIVP$.
 \end{enumerate}
 \end{Th}
\begin{Th}{\rm (D.~Banaszewski~\cite{DB})}
 \begin{enumerate}
 \item[(1)]
 There exists $f\in\Add\cap\PR$ such that $f\not\in\CIVP\cup\D$.
\item[(2)]
 There exists $f\in\Add\cap\ACS$ such that $f\not\in\PR$.
 \item[(3)]
 There exists $f\in\Add\cap\CIVP$ such that $f\not\in\D$.
 \item[(4)]
 There exists $f\in\Add\cap\D$ such that $f\not\in\Conn$.
 \end{enumerate}
 \end{Th}
 D.~Banaszewski posed also the following {\bf open question}.
 \begin{Qu}
 Does there exist $f\in\Add\cap\Conn\setminus\ACS$?
 \end{Qu}
 Moreover, we are unable to construct a discontinuous function
 $f\in\Add\cap\Ext$. Note that it is easy to construct a discontinuous
function $f\in\Add\cap\SCIVP\cap\ACS$.  (This holds because there exists a
Hamel base which contains a perfect set.)


 \section{Darboux like functions versus quasi-continuity}
\newcommand{\ACH}{{\rm ACH}}
\newcommand{\QC}{{\rm QC}}
\newcommand{\Cliq}{{\rm Cliq}}
\newcommand{\CT}{{\rm CT}}

We now give some facts that are related to a different kind of
 discontinuity. In this investigation we will also discuss some
 relations with the previous classes of functions.
Recall the following notions.

Let $f:X\rightarrow Y$ be a function. Then:
\begin{description}
\item[ACH]
-- $f$ is an {\it almost continuous function in the sense of
 Husain}, if for every $x\in X$ and for each open neighborhood
 $V$ of $f(x)$ in $Y$, $\cl(f^{-1}(V))$ is a neighborhood of $x$.
\item[CT]
 -- $f$ is said to be of the {\it Cesaro type} if there exist
 non-empty open sets $U$ and $V$ in $X$ and $Y$, respectively,
 such that $U\subset\cl(f^{-1}(y))$ for all $y\in V$.
\item[QC]
 -- $f$ is said to be {\it quasi-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 non-empty open subset
 $W\subset U$ such that $f(W)\subset V$.
\item[CLIQ]
-- Let $Y$ be a metric space with metric $\varrho$.
 Then $f$ is {\it cliquish} if for every $x\in X$, for each open
 neighborhood $U$ of $x$ and for every $\varepsilon>0$ there
 exists a non-empty open subset $W\subset U$ such that
$\varrho(f(y),f(z))<\varepsilon$, for all $y,z\in W$.
\end{description}

T. Husain defined the notion of almost continuous functions in
 the sense of Husain in~\cite{43}. Note that the function
 $f\colon [0,1]\to\mathR$
 defined by $f(x)=\sin (\frac 1x)$ for $x>0$, and $f(0)=0$ is a
 Darboux function of Baire class 1 but is not almost continuous
 in the sense of Husain. Thus this type of almost continuity is
 different from the other in a very restrictive class of
 functions. It should be noted that almost
 continuity in the sense of Husain was defined earlier by
 H.~Blumberg~\cite{4}, who
used the phrase ``densely approached''. S.~Kempisty defined
the notion of quasi continuous function in~\cite{SK}. (See also \cite{57}.)
Finally, the notion of cliquishness was introduced by H.~P.~Thielman
in~\cite{Thi}.

Clearly, each function with values in a metric space, which is
 quasicontinuous, is cliquish. Moreover, it is worth to notice
 that for the real functions defined on a Baire space,
\begin{itemize}
\item
 $f\in\QC$ iff the restriction $f\restr C(f)$, of $f$ to the set
 of all points at which $f$ is continuous, is dense in $f$;
\item
 $f\in\Cliq$ iff $f$ is pointwise discontinuous, i.e., the set
 $C(f)$ is dense in $X$.
\item
Each $f\in\Cliq$ has the Baire property.
\end{itemize}

Also, there exist quasi-continuous functions
 $f\colon\mathR\to\mathR$ that are not almost continuous in the sense of
Husain nor in the sense of Stallings.

Darboux like functions in the class of quasi-continuous functions
 were studied in two papers, by
 R.~Gibson and I.~Rec{\l}aw in~\cite{37}, and independently, by
 T.~Natkaniec in \cite{63}.

\begin{Th}{\rm (Gibson, Rec{\l}aw~\cite{37})}
\begin{enumerate}
\item[(1)]
 There exists $f\colon {\mathI}\to {\mathI}$ such that
 $f\in\PR\cap\QC$ but $f\notin\D$.
\item[(2)]
 There exists $f\colon {\mathI}\to {\mathI}$ such that
 $f\in\PR\cap\Cliq$ but $f\notin\QC$.
\item[(3)]
There exists $f\colon {\mathI}\to {\mathI}$ such that
 $f\in\QC\cap\D$ but $f\notin\Conn$.
\item[(4)]
If $f\colon\mathR\to\mathR$ and $f\in\QC$, then $f\in\PR$ iff $f\in\PC$.
\item[(5)]
There exists $f\colon {\mathI}\to {\mathI}$ such that
 $f\in\Cliq\cap\PC$ but $f\notin\PR$.
\end{enumerate}
\end{Th}

Gibson and Rec{\l}aw also asked the questions, for functions
 $f\colon\mathR\to\mathR$,

\begin{Qu}
Does $\QC\cap\Conn\subset\ACS$?
\end{Qu}

This question is answered in the negative by A.~Andryszczak (Nowik) and
M.~Szysz\-kow\-ski. (See~\cite{37}.) They observed that the function $f$
constructed in \cite{45}  by J.~Jastrz{\c{e}}bski has the property that
$f\in\QC\cap\Conn$ but $f\notin\ACS$. (See also~\cite{63}.)

\begin{Qu}
Does $\QC\cap\ACS\subset\Ext$?
\end{Qu}

To answer this question, we prove the following
\begin{Th}
 There exists a quasi-continuous function $f\colon {\mathI}\to
 {\mathI}$ in the class $\ACS\setminus\CIVP$.
 \end{Th}
 \pf
 Let $C$ be the ternary Cantor set and let $(I_{n,m})_{n,m}$ be
 the sequence of all components of ${\mathI}\setminus C$ such that
 \begin{itemize}
 \item
 for each $n$, $\bigcup_mI_{n,m}$ is dense in $C$.
 \end{itemize}
 Let $(q_n)_n$ be a sequence of all rationals. Moreover, let
 $C_0=C\setminus \bigcup_{n,m}\cl(I_{n,m})$ and let $B\subset
 C_0$ be a Bernstein set in $C$, i.e., $B\cap P\neq\emptyset\neq
 C\setminus P$ for each non-empty perfect set $P\subset C$.

Now let $(F_{\alpha})_{\alpha<\co}$ be a sequence of all minimal
 blocking sets in ${\mathI}\times {\mathI}$ such that
 $\dom(F_{\alpha})\cap
 C_0\neq\emptyset$. Note that $|\dom(F_{\alpha})\cap B|=\co$
 for each $\alpha<\co$. For each $\alpha<\co$ choose
 $(x_{\alpha},y_{\alpha})\in F_{\alpha}$ such that
 \begin{itemize}
 \item
 $x_{\alpha}\in B$;
 \item
 $x_{\alpha}\neq x_{\beta}$ for $\alpha\neq\beta$.
 \end{itemize}

Define $f\colon {\mathI}\to {\mathI}$ by
 $$f(x)=\left \{ \begin{array}{ll}
 q_n & \mbox{for $x\in\bigcup_m\cl(I_{n,m})$;}\\
 y_{\alpha} & \mbox{for $x=x_{\alpha}$, $\alpha<\co$;}\\
 0 & \mbox{otherwise.}
 \end{array}\right. $$
 Observe that $f_0=f\restr\bigcup_{n,m}I_{n,m}$ is continuous and
 $f_0$ is dense in $f$. Thus $f$ is quasi-continuous.

Now, $f$ is almost continuous. Indeed, if $F\subset {\mathI}^2$ is a
 minimal blocking set then either $\dom(F)\subset\cl(I_{n,m})$
 for some $n,m\in N$ or $F=F_{\alpha}$ for some $\alpha<\co$.
 (See, e.g.,~\cite{60}.)
 Therefore either $(x,q_n)\in F$ for some $x$ (because
 $\rng(F)=\mathR$) or $(x_{\alpha},y_{\alpha})\in F\cap f$. Thus
 $f\cap F\neq\emptyset$.

Finally, let $K$ be a Cantor set such that $K\subset {\mathI}\setminus
 \mathQ$. Then $f^{-1}(K)\subset B$, so it contains no Cantor set.
 \Qed

\begin{Co}
 $\QC\cap\ACS\setminus\Ext\neq\emptyset$.
\end{Co}

\subsection{Decomposition of the continuity}

\begin{Th}\label{Sm}
{\rm (D.~B.~Smith~\cite{76})}
 $f:[a,b]\to\mathR$ is continuous if and only if it satisfies the
 conjunction of the following three conditions:
\begin{enumerate}
\item[(1)]
$f\in\ACS$;
\item[(2)]
$f\in\ACH$;
\item[(3)]
$f\not\in\CT$.
\end{enumerate}
\end{Th}

With the examples given in the paper~\cite{76} and the examples
 given by R.~J.~Fleissner~\cite{25} and by J.~Brown~\cite{10} it
 follows that the three conditions are not redundant. At a real
 variable conference at Auburn University and at the XV~Summer
 Symposium in Real Analysis held in Smolenice Castle, Smolenice,
 Czechoslovakia, August 1991, R.~Gibson answered the following
 three questions. (Remember Chart~1! See page \pageref{chartone})

\begin{Qu}
In Theorem~\ref{Sm}, if $f\in\ACS$ is replaced with
the stronger condition $f\in\Ext$, are the three conditions redundant?
\end{Qu}
Answer: {\bf No.} See~\cite{28}.

\begin{Qu}
 In Theorem~\ref{Sm}, if the condition $f\in\ACS$ is replaced
 with the weaker condition $f\in\D$, is the theorem
true?
\end{Qu}
Answer: {\bf Yes.} See~\cite{28}.

\medskip
In~\cite{28}, it is shown that if $f:[a,b]\to\mathR$ is almost
 continuous in the sense of Husain, then $f$ is peripherally
 continuous. Thus it follows
 that we can weaken condition (1) of the theorem of B.~D.~Smith
 to include all Darboux functions, but we can not weaken
 condition (1) to include all peripherally continuous functions.

\medskip
In \cite{75}, Sm{\'\i}tal and Stanova proved the following theorem.
\begin{Th}
{\rm (Sm{\'\i}tal, Stanova~\cite{75})}
 There exists a function $h\colon\mathR\to\mathR$ such that
 $h\in\ACS,$ $h\notin\CT$ and $h\notin\QC$.
\end{Th}
This suggests the following problem.
\begin{Qu}
 Does there exists a function $h\colon\mathR\to\mathR$ such that
 $h\in\Ext$, $h\notin\CT$, and $h\notin\QC$?
\end{Qu}
Answer: {\bf Yes.} Recently H.~Rosen remarked that the Croft's function,
i.e., Darboux Baire~1 function $h\colon\mathR\to\mathR$ that equals $0$
almost everywhere (See~\cite{12}.) belongs to the class $\Ext\setminus
\CT\cup\QC$.

J.~Sm{\'\i}tal and E.~Stanova~\cite{75} gave a generalization of
 the theorem of Smith by proving the following theorem.
\begin{Th}\label{SS}
{\rm (Sm{\'\i}tal, Stanova~\cite{75})}
Let $X$ be a $T_3$ locally connected Baire topological space. A
 function $f:X\to\mathR$ is continuous if and only if it
 satisfies the conjunction of the following three conditions:
\begin{enumerate}
\item[(1)]
$f\in\ACS$;
\item[(2)]
$f\in\ACH$;
\item[(3)]
$f\not\in\CT$.
\end{enumerate}
\end{Th}

\begin{Qu}
 Is Theorem~\ref{SS} true when condition (1) is replaced with the
 condition $f\in\D$?
\end{Qu}
Answer: {\bf Yes. } See~\cite{28}.


\section{Some characterizations of Darboux like functions}
\subsection{Extendability and peripheral families}
The following question was posed in~\cite{26}.

\begin{Qu}
Is there a ''nice condition'' that characterizes extendable functions?
\end{Qu}
See also Question~\ref{q1}.
 Concerning this question, Gibson and Roush in~\cite{35} defined
 a family of peripheral intervals for a function $f\colon
 {\mathI}\to {\mathI}$.

\medskip
\noindent
 {\bf Definition. }Let $f\colon {\mathI}\to {\mathI}$ be a
 function. A family of {\it peripheral intervals}
 ({\bf PI}) for $f$ consists of a sequence of ordered pairs
 $(I_n,J_n)$ of subintervals of ${\mathI}$ such that
\begin{enumerate}
\item[(1)]
 $I_n$ is open in ${\mathI}$ and the length of $I_n$ converges to $0$;
\item[(2)]
for each $x\in {\mathI}$ and for any $\varepsilon >0$ there exists
 $(I_n,J_n)$ such that $x\in I_n$, $f(x)\in J_n$, and the length
 of $I_n$ and $J_n$ are less than $\varepsilon$;
\item[(3)]
 both endpoints of $I_n$ map into $J_n$;
 \item[(4)]
  if $I_n$ and $I_m$ have points in common but neither is
a subset of the other, then $J_n$ and $J_m$ have points in common.
\end{enumerate}

Then in the same paper~\cite{35}, Gibson and Roush proved the following
two theorems.

\begin{Th}\label{pi}
{\rm (Gibson, Roush~\cite{35})}
 Assume that for $f\colon {\mathI}\to {\mathI}$ there exists a
 family of $\PI$. Then $f$ is the restriction of a connectivity
 function $F\colon {\mathI}^2\to {\mathI}$ such that $F$ is
 continuous on the complement of ${\mathI}\times \{0\}$, where
 ${\mathI}$ is embedded in ${\mathI}^2$ as ${\mathI}\times
 \{0\}$.
\end{Th}

Note also that a function $F$ in Theorem~\ref{pi} can be chosen
 to be constant on intervals $\{ 0\}\times {\mathI}$ and $\{
 1\}\times {\mathI}$.

\begin{Th}\label{pii}
{\rm (Gibson, Roush~\cite{35})}
 The existence of a family of $\PI$ is both necessary and
 sufficient that a function $f\colon {\mathI}\to {\mathI}$ be an
 extendable function.
\end{Th}

It should be noted that Theorems \ref{pi} and \ref{pii} gives a
 result that is a generalization of Tietze's extension theorem
 for closed set ${\mathI}\times \{ 0\}$ in ${\mathI}^2$ and for
an extendable function defined on ${\mathI}\times \{ 0\}$. The
 definition of a family of $\PI$ is long and difficult to deal
 %ch
with\footnote{Nevertheless it can be useful. See~\cite{JN}.}. Thus can this
definition of a family of $\PI$ be replaced
 with a ``nice condition''?

\subsection{Negligible sets}
Assume that $\cal K$ is a class of functions from $X$ into $Y$ and
$g\in{\cal K}$. A set $M\subset X$ is called {\it $g$-negligible} with
respect to $\cal K$, if every function $f\colon X\to Y$ which agrees with
$g$ on $X\setminus M$ belongs to $\cal K$.

In 1970, J.~Brown proved the following result.
\begin{Th}{\rm (Brown~\cite{7})}
If ${\cal K}=\Conn$ and $g\in {\mathI}^{\mathI}\cap{\cal K}$  then the
following statements are equivalent:
\begin{enumerate}
\item[(i)]
$g$ is dense in ${\mathI}^2$;
\item[(ii)]
 every nowhere dense subset of ${\mathI}$ is $g$-negligible with
 respect to $\cal K$;
\item[(iii)]
 there exists a dense subset of ${\mathI}$ which is
 $g$-negligible with respect to $\cal K$.
\end{enumerate}
\end{Th}

In \cite{51} K.~Kellum showed that Brown's characterization is
 still valid when $\cal K$ is replaced by the class $\ACS$.
A stronger result for the class $\Ext$ was obtained recently by H.~Rosen.

\begin{Th}\label{negli}
{\rm (Rosen~\cite{67})}
 If ${\cal K}=\Ext$ and $g\in {\mathI}^{\mathI}\cap{\cal K}$ then
 the following statements are equivalent:
\begin{enumerate}
\item[(i)]
$g$ is dense in ${\mathI}^2$;
\item[(ii)]
 every nowhere dense subset of ${\mathI}$ is $g$-negligible with
 respect to $\cal K$;
\item[(iii)]
 there exists a dense $G_{\delta}$ subset of ${\mathI}$ which is
 $g$-negligible with respect to $\cal K$.
\end{enumerate}
\end{Th}
 Note that the analogous result holds also in the class of all
 real functions defined on $\mathR$~\cite{18,71}.
 Theorem~\ref{negli} together with examples of extendable
 functions that are dense in ${\mathI}^2$ (See~\cite{8,10}.) and
 extendable functions that are dense in
 $\mathR^2$~(See~\cite{18,71}.) are the useful instruments to
 construct extendable functions. (See~\cite{ima,18,64,71,72}.)



\subsection{Darboux like functions that are characterizable by
 images, preimages and associated sets}

Recall the following definitions.
For the families $\A,\B\subset\P(\mathR)$ we define
\begin{align*}
\CAB &=\{ f\in\RRR\colon(\forall A\in\A)\, (f(A)\in\B)\},\\
\intertext{and}
\DAB &=\{ f\in\RRR\colon(\forall B\in\B)\,(f^{-1}(B)\in\A)\}.
\end{align*}

Also, we say that a family $\F$ of real functions is
\begin{itemize}
\item
{\em characterizable by images}\/ of sets
 if $\F=\CAB$ for some $\A,\B\subset\P(\mathR)$;
\item
{\em characterizable by preimages}\/ of sets if
$\F=\DAB$ for some $\A,\B\subset\P(\mathR)$;
\item
{\em topologized}\/ if $\F=\DAB$ for some
\underline{topologies} $\A,\B$ on $\mathR$;
 \item
 {\em characterizable by associated sets}\/ if there
exists an $\A\subset\P(\mathR)$ such that
\begin{quote}
$f\in\F$ if and only if for every
$\alpha\in\mathR$, the ``associated'' sets
$E^{\alpha}(f)=\{ x\colon f(x)<\alpha\}$ and
$E_{\alpha}(f)=\{ x\colon f(x)>\alpha\}$ belong to $\A$.
\end{quote}
 (i.e., $\F=\DAB$ for $\B=\{ (a,\infty)\colon a\in\mathR\}\cup \{
 (-\infty, a)\colon a\in\mathR\}$.)
\end{itemize}

Clearly the class $\CC$ can be defined by preimages of
open sets, so it can be topologized and characterized
by associated sets.
On the other hand, this class cannot be
characterized by images of sets \cite{Ve}.

The problem of characterizing some Darboux like functions by
 associated sets has been studied in several papers.

\begin{Th}\mbox{}

\begin{itemize}
\item
 The class $\D$ cannot be characterized by associated sets {\rm
 (Bruckner~\cite{13})}.
\item
 The class $\Conn$ cannot be characterized by associated sets
 {\rm (Cristian, Tevy~\cite{21})}.
\item
 The class $\ACS$ cannot be characterized by associated sets {\rm
 (Kellum~\cite{51})}.
\item
 The class $\Ext$ cannot be characterized by associated sets {\rm
 (Rosen~\cite{72})}.
\end{itemize}
\end{Th}

The problem of characterizing Darboux like functions by images
 and by preimages of sets has been recently addressed by
 Ciesielski and Natkaniec.

\begin{Th}{\rm (Ciesielski, Natkaniec~\cite{ima})}
\begin{enumerate}
\item[(1)]
 The classes: $\Ext$, $\ACS$,
 %ch
 $\ACS\cap\PR$,
 $\Conn$, $\D$, $\SCIVP$, $\CIVP$, and $\WCIVP$ cannot be characterized by
preimages
of sets.
\item[(2)]
 The classes: $\PR$ and $\PC$ can be characterized
by preimages of sets as $\CAB$ with $\B$ being the natural
 topology on $\mathR$. However, they can neither be topologized
 nor be characterized by associated sets.
\end{enumerate}
\end{Th}

\begin{Th}{\rm (Ciesielski, Natkaniec~\cite{ima})}
\begin{enumerate}
\item[(1)]
 The classes: $\Ext$, $\ACS$,
 %ch
 $\ACS\cap\PR$,
  $\Conn$, $\SCIVP$, $\PR$, and $\PC$ cannot be characterized by images of
sets.
\item[(2)]
 The classes: $\D$, $\CIVP$ and $\WCIVP$ can be characterized by
 images of sets.
\end{enumerate}
\end{Th}

We can complete those results and determine whether the classes
 $\ACS\cap\PR$ and $\PB$ can be characterized by images or by
 preimages of sets.

\begin{Th}\mbox{}

\begin{enumerate}
\item[(1)]
 The class $\ACS\cap\PR$ can neither be characterized by images
 nor by preimages of sets.
\item[(2)]
 The class $\PB$ cannot be characterized by images. It can be
 characterized by preimages, however cannot be topologized nor
 characterized by associated sets.
\end{enumerate}
\end{Th}
\pf
 It is known that if $\F\subset\RRR$ is characterizable by images
 and $\Ext\subset\F\subset\D$, then $\F=\D$ \cite[Theorem
 4]{ima}. Thus, $\ACS\cap\PR$ cannot be characterized by images.

Similarly, it is known that if $\F\subset\RRR$ is characterizable
 by preimages and $\Ext\subset\F$, then $\F\not\subset\D$
 \cite[Theorem 5]{ima}. Thus, $\ACS\cap\PR$ cannot be
 characterized by preimages.

Also, it is known that if $\F\in\RRR$ satisfies the following conditions
\begin{enumerate}
 \item[(1)]
 $\Const\subset\F$;
 \item[(2)]
 for every distinct $a,b\in\mathR$ there exists $f\in\F$
 with $f(\mathR)=\{ f(a),f(b)\}\in [\mathR]^2$;
 \item[(3)]
 there exists $Z\subset\mathR$ such that any distinct $a,b\in\mathR$
the  ``characteristic'' function
 $$\varphi^Z_{a,b}=\left\{
 \begin{array}{ll}
 a &\mbox{if $x\in Z$,}\\
 b &\mbox{if $x\not\in Z$}
 \end{array}\right.$$
does not belong to $\F$,
 \end{enumerate}
 then $\F$ cannot be characterized by images of sets.
 (See~\cite[Corollary 1]{ima}.) Therefore the class $\PB$ cannot
 be characterized by images.

However, $\PB=\DAB$, where
\begin{itemize}
\item
 $A\in\A$ iff for each interval ${\mathI}\subset\mathR$, if
 $|A\cap {\mathI}|>\omega$ then $A\cap {\mathI}$ contains a
 non-empty perfect set;
\item
$\B$ is the natural topology on $\mathR$
\end{itemize}

To see that $\PB$ cannot be topologized recall that if $\F$ is
 topologized and $\PR\subset\F$ then $\F=\RRR$. (See
 \cite[Corollary 3]{ima}.)

Finally, we shall prove that $\PB$ cannot be characterized by
 associated sets. Assume that $\PR\subset\F$ and $\F$ can be
 characterized by associated sets. We will prove that
 $\F\setminus\PB\neq\emptyset$.

Let $\A$ denote the family of all associated sets of $\F$
and let $C$ be the ternary Cantor set.
 Divide the set $\mathR\setminus C$ onto two sets $A_0$ and
 $A_1$, each Cantor dense in $\mathR$. Let $C_1$ be a subset of
 $C$ such that $|C_1|=\co$ and $C_1$ contains no Cantor set. Put
 $C_0=C\setminus C_1$.
Since the characteristic functions
$\charf{A_0\cup C_0},\charf{A_1}\in\PR\subset\F$,
 the sets $A_0\cup C_0$, $A_1\cup C_1$, $A_1$, and $A_0\cup C$
 belong to $\A$.
Then $f=\charf{A_0\cup C_0}-\charf{A_1}\in\F$.
 However, $f\not\in\PB$, because the set $f^{-1}(-1,1)=C_1$ is of
 the size $\co$ and contains no Cantor set.
\Qed

\section{Darboux like functions of $n$ variables}
\newcommand{\QCOMP}{{\rm QCOMP}}

First, we should notice that for $n>1$, Chart~1 is no longer valid.
Many of the results that are presented in this survey
follow from the following fact.
\begin{Th}\label{t2}
{\rm (Hagan~\cite{39}, Whyburn~\cite{81})}
If $f\colon {\mathI}^n\to {\mathI}$ and $n>1$, then $\Conn=\PC$.
\end{Th}

From the paper~\cite{77} by Stallings, it
follows the following inclusions
\begin{Th}
{\rm (Stallings~\cite{77})} Assume that $n>1$.
\begin{enumerate}
\item[(1)]
 If $f\colon {\mathI}^n\to {\mathI}$ is a connectivity function,
 then $f$ is almost continuous in the sense
of Stallings.
\item[(2)]
 If $f\colon {\mathI}^n\to {\mathI}$ is a connectivity function,
 then $f$ is a Darboux function.
\end{enumerate}
\end{Th}
 Thus in the class of real functions of $n$ variables (for $n>1$)
 the following inclusions hold
$$\Ext\subset\Conn=\PC\subset\D\cap\ACS.$$

The examples showing that $\D\not\subset\ACS$ and
 $\ACS\not\subset\D$ can be found in \cite[Examples 1.6 and
 1.7]{60}. An example of $f\colon {\mathI}^2\to {\mathI}$ such
 that $f\in \ACS\cap\D\setminus\Conn$ is constructed in
 \cite[Example 1]{68}.
The following
{\bf open} problem is posed by K.~Ciesielski and J.~Wojciechowski~\cite{CW}.
\begin{Qu}
 Is the inclusion $\Ext\subset\Conn$ proper in the class of all
 real functions of $n$ variables, when $n>1$?
\end{Qu}

From the paper~\cite{36}, by Gibson, Rosen and Roush we have that
 if $n>1$ and $f\colon {\mathI}^n\to {\mathI}$ is a connectivity
 function, then for any $x\in {\mathI}^n$ and for any line
 segment $L$ or a union of two line segment $L_1$ and $L_2$
 containing $x$ as a limit point from two directions, there
 exists a perfect set $P$ containing $x$ as a limit point from
 two direction such that $f\restr P$ is continuous. Thus we can
 say that if $f\colon {\mathI}^n\to {\mathI}$ is a connectivity
 function, then $f$ has a perfect road.

%ch
A strengthening of this  result have been obtained recently by
K.~Ciesielski and J.~Wojciechowski. (Note that it also implies that $\PC$
functions on $\mathR^2$ have a two-dimensional version of $\SCIVP$.)
\begin{Th}
{\rm (Ciesielski, Wojciechowski~\cite{CW})}
 Let $n>1$ and $g\colon\mathR^n\to\mathR$ be peripherally
 continuous. If $X$ is a non-empty
connected perfect subset of $\mathR^n$, then there exists a
non-empty
perfect subset $P$ of $X$ such that the restriction $g\restr P$
is continuous.
\end{Th}

It is a well-known fact that, if $f\colon {\mathI}^n\to
 {\mathI}$, $n>1$, is continuous and $z$ separates the range of
 $f$, then $f^{-1}(z)$ separates ${\mathI}^n$. From the
 paper~\cite{68}, by Rosen, Gibson, and Roush it follows that the
 same conclusion holds for a connectivity function $f\colon
 {\mathI}^n\to {\mathI}$, $n>1$. However, this is not true for
 Darboux functions nor for almost continuous functions in the
sense of Stallings,~\cite{77}.

In particular, Rosen, Gibson and Roush proved in~\cite{68} that
 if $f\colon {\mathI}^2\to {\mathI}$ is a connectivity function
 and $z$ is an interior point of $f({\mathI}^2)$, then any point
 of $f^{-1}([0,z))$ and any point of $f^{-1}((z,1])$ lie in
 different quasi-components of ${\mathI}^2\setminus f^{-1}(z)$.
 They gave an example that shows that this conclusion is false
 for Darboux functions.

In~\cite{69} H.~Rosen proved: if for all $z\in f({\mathI}^2)$,
 any point of $f^{-1}([0,z))$ and any point of $f^{-1}((z,1])$
 lie in different quasi-components of ${\mathI}^2\setminus
 f^{-1}(z)$, then $f$ is a Darboux function. In Theorem~1
 of~\cite{69}, H. Rosen (using results from~\cite{80*}) proved
 that if $f\colon {\mathI}^n\to {\mathI}$, $n>1$, and $f\in\PC$,
 then the quasi-components and the components of $f^{-1}(z)$ are
 the same.

For future work we make the following definition.
\begin{description}
\item[QCOMP]
 -- $f\colon {\mathI}^2\to {\mathI}$ has the {\it $\QCOMP$
 property} if for every point $z\in f({\mathI}^2)$, any point of
 $f^{-1}([0,z))$ and any point of $f^{-1}((z,1])$ lie in
 different quasi-component.
\end{description}

Clearly, if $f\colon {\mathI}^2\to {\mathI}$, then
$$\Conn=\PC\subset\QCOMP\subset D.$$

Define the function $f\colon [-1,1]\times [0,1]\to [-1,1]$ as follows:
$$
f(x,y)=\left\{
\begin{array}{cc}
\sin (\frac 1y) & \text{if $y>0$}\\
0 & \text{if $y=0$.}
\end{array}\right.
$$
 Clearly, the quasi-components and the components of the
 complement of $f^{-1}(z)$, for all $z\in [-1,1]$, are the same.
 Hence $f\in\QCOMP\subset\D$. Also $f\in\ACS$.

However $f\notin\Conn$.
Indeed, fix $y_k\in (0,1]$ such that $y_{k-1}>y_k$ and $\sin
(\frac 1{y_k})=1$ for $k=1,2,3,....$ Let $A=(
\bigcup _k([-1,1]\times
 \{y_k\}))\cup (\{0\}\times [0,1])\cup \{(1,0)\}$. Then $A$ is
 connected but $f\restr A$ is not connected.

Now, define the function $g\colon[-1,1]\times [0,1]\to [-1,1]$ as
 follows:
$$
g(x,y)=\left\{
\begin{array}{cc}
\sin (\frac 1y) & \text{if $y>0$}\\
x & \text{if $y=0$.}
\end{array}\right.
$$
Clearly, the quasi-components of the complement of $g^{-1}(0)$
are not connected. Thus $g\notin \Conn=\PC$. However, $g\in
\D\cap \ACS$. Thus, $\D\cap\ACS\setminus\QCOMP\neq\emptyset$.

This suggests the following {\bf open question}.
\begin{Qu}
If $f\colon {\mathI}^n\to {\mathI}$, $n>1$, and
$f\in\QCOMP$, is $f\in\ACS$?
\end{Qu}

\section{Algebraic operations}
\subsection{Compositions}
 At the 11th Summer Symposium on Real Analysis at Esztergom,
 Hungary, R.~Gibson asked the following question.
\begin{Qu}
If $f,g\in\Ext$, is the composition $g\circ f\in\Ext$?
\end{Qu}
This question remains {\bf open}.

Obviously, if $h\colon X\to Y$ is the composition of connectivity
 functions, then $h$ must be a Darboux function.
On the other hand, we have the following.
\begin{Th}
\mbox{}
\begin{enumerate}
\item[(1)]
 There exist almost continuous functions $f\colon {\mathI}^n\to
 {\mathI}^m$ and $g\colon {\mathI}^m\to {\mathI}^n$ such that
 $g\circ f$ has no
fixed point and is not almost continuous. {\rm
(Kellum~\cite{50*})}
\item[(2)]
 There exists almost continuous function $f\colon\mathR\to\mathR$
 such that $f\circ f$ is not almost continuous. {\rm
(Kellum~\cite{sar})}
\end{enumerate}
\end{Th}

This suggests the following question. (See~\cite{62} or~\cite{GMN}.)
\begin{Qu}\label{comp}
Is every Darboux function the composition of two (finite many) of
$\ACS$ (or $\Conn$) functions?
\end{Qu}

Note that the classes $\SCIVP$ and $\CIVP$ are closed with
 respect to compositions, so no $f\in\D\setminus\SCIVP$ can be
 expressed as the composition of extendable functions. In
 particular, by Theorem~\ref{c5}, there are almost continuous
 functions that cannot be written as the composition of
 extendable functions.

For functions defined on the plane, Question~\ref{comp} was
 solved by H.~Rosen.
\begin{Th}{\rm (Rosen~\cite{69})}
There exists
 a Darboux function $h\colon {\mathI}^2\to {\mathI}$ that is also
 an almost continuous function, that is not the composite of any
 two connectivity functions $f\colon {\mathI}^2\to {\mathI}$ and
 $g\colon {\mathI}\to {\mathI}$.
\end{Th}

Generally, for real functions defined on $\mathR$,
 Question~\ref{comp} remains {\bf open}. However, there are some
 partial results in this direction.
\begin{Th}{\rm (Natkaniec~\cite{62})}
Assume that the covering of category is equal to continuum.
 Then every function with dense level sets can be expressed as
 the composition of two $\ACS$ functions.
\end{Th}
We do not know whether this theorem can be proved in ZFC.
Question~\ref{comp} has a surprising solution in the class $\Add$
 of additive function. In fact, D.~Banaszewski proved recently
 the following theorem.
\begin{Th}{\rm (D.~Banaszewski~\cite{DB})}\label{addcom}
Assume that the covering of category is equal to continuum
and $f\colon\mathR\to\mathR$ is an additive function.
Then the following are equivalent:
\begin{enumerate}
\item[(i)]
${\rm dim}({\rm ker}(f))\neq 1$;
\item[(ii)]
$f$ is the composition of two $\ACS$ additive functions;
 \item[(iii)]
$f$ is the composition of two $\Conn$ additive functions.
\end{enumerate}
\end{Th}
 Moreover, K.~Ciesielski observed that Theorem~\ref{addcom} can
 be proved in ZFC, without extra set-theoretic assumptions.
(See~\cite{banan}.)

\begin{Th}
{\rm (K.~Banaszewski~\cite{KB})}
 There exists a $\PR$-function $h\colon\mathR\to\mathR$ with the
 following property:
\begin{quote}
 for every $f\colon\mathR\to\mathR$ there exists $f^{\ast}\in\PR$
 such that $f=f^{\ast}\circ h$.
\end{quote}
 In particular, every real function can be expressed as the
 composition of two $\PR$-functions.
\end{Th}

\subsection{Pointwise limits}
\begin{Th}
 Any real-valued function defined on an interval is the pointwise
 limit of a sequence
\begin{itemize}
\item
of Darboux functions {\rm (Lindenbaum~\cite{55})};
\item
 of $\Conn$ functions {\rm (Phillips~\cite{66})};
\item
of $\ACS$ functions {\rm (Kellum~\cite{55})};
\item
of $\CIVP\cap\D$ functions {\rm (K.~Banaszewski~\cite{2})};
\item
of $\Ext$ functions {\rm (Rosen~\cite{67})}.
\end{itemize}
\end{Th}

\subsection{Uniform limits}
\newcommand{\UL}{{\rm UL}}

In this subsection we shall deal with the classes of uniform
 limits of sequences of Darboux like functions. Let
 $\overline{\F}$ denote the uniform limit of
a class $\F$.
\begin{Th}
In the class of all functions from $\mathR$ into $\mathR$,
\begin{enumerate}
\item[(1)]
The class $\D$ is not closed with respect to uniform limits.
 {\rm (Sier\-pi{\'n}\-ski~\cite{WS})}
\item[(2)]
 There exists $f\in\overline{\ACS}\setminus\D$. Thus the classes
 $\ACS$ and $\Conn$ are not closed with respect to uniform
 limits. {\rm (Kellum~\cite{55})}
\item[(3)]
The class $\PC$ is closed with respect to uniform limits.
 {\rm (Gi\-b\-son, Roush~\cite{32})}
\item[(4)]
The class $\PR$ is closed with respect to uniform limits.
 {\rm (K.~Ba\-na\-szew\-ski~\cite{KB1})}
\item[(4)]
 The class $\CIVP$ is not closed with respect to uniform
 limits. {\rm (K.~Ba\-na\-szew\-ski~\cite{2})}
\item[(5)]
 The class $\Ext$ is not closed with respect to uniform
 limits. {\rm (Rosen~\cite{70})}
\end{enumerate}
\end{Th}

Moreover, we have the following examples.
\begin{Th}
In the class of all functions from $\mathR$ into $\mathR$,
\begin{enumerate}
\item[(1)]
 There exists $f\in\D\setminus\overline{\Conn}$. {\rm (Gibson,
 Roush~\cite{32})}
\item[(2)]
 There exists $f\in\Conn\setminus\overline{\ACS}$. {\rm
 (Jastrz{\c{e}}bski~\cite{45})}
\item[(3)]
There exists $f\in\ACS\setminus\overline{\Ext}$. {\rm (Rosen~\cite{72})}
\end{enumerate}
\end{Th}

The class {\bf UL} of all uniform limits of sequences of Darboux
 functions has been described by A.~Bruckner, J.~Ceder and
 M.~Weiss~\cite{16}.
Note that the following inclusions hold:
$$\D\cup\CIVP\subset\UL\subset\PC.$$

The uniform closures of the classes $\CIVP$ and $\SCIVP$ are
 described by K.~Banaszewski.
\begin{Th}\mbox{}
\begin{enumerate}
\item[(1)]
$\overline{\CIVP}=\PR\cap\UL=\WCIVP\cap\UL$ {\rm (K.~Banaszewski~\cite{2})}
\item[(2)]
 $\overline{\SCIVP}=\overline{\CIVP}$ {\rm (K.~Banaszewski,
 T.~Natkaniec~\cite{3})}
\end{enumerate}
\end{Th}
\begin{Co}
There is $f\in\PR\cap\UL\setminus\D$.
\end{Co}
The answer to the following question is {\bf unknown}.
\begin{Qu}
Does there exist $f\in\ACS\cap\PR\setminus\overline{\Ext}$?
\end{Qu}

The following problems also remain {\bf open}.
\begin{Qu}\label{uni}
 Characterize the uniform limits of sequences of $\Ext$
functions ($\ACS$ functions, $\Conn$ functions or $\WCIVP$ functions).
\end{Qu}

Note that a partial solution of Question~\ref{uni} for the class
 $\ACS$ is contained in the paper by T.~Natkaniec~\cite{60}.
In~\cite{32}, Gibson and Roush proved the following
theorems.
\begin{Th}\label{ul}
{\rm (Gibson, Roush~\cite{32})}
 Let $X$ be a metric space. Then the uniform limit $f$ of a
 sequence $f_m\colon X\to\mathR$ of peripherally continuous
 functions is peripherally continuous.
\end{Th}
 As a corollary to Theorem~\ref{ul} we have the
following result.
\begin{Co}
 Let $(f_m)$ be a sequence of functions from ${\mathI}^n$ into
 ${\mathI}$, where $n>1$. If each $f_m$ is a connectivity
 function and $f_m$ converges to $f$ uniformly, then $f$ is a
 connectivity function.
\end{Co}

\subsection{Sums}
It is known that every function $f\colon\mathR\to\mathR$ is the sum of two:
\begin{itemize}
\item
Darboux functions (Lindenbaum~\cite{55});
\item
connectivity functions (Phillips~\cite{66});
\item
almost continuous functions (Kellum~\cite{49});
\item
perfect road functions (K.~Banaszewski~\cite{KB1});
\item
peripherally continuous functions (K.~Banaszewski~\cite{KB1}).
\end{itemize}

At the 11th Summer Symposium on Real Analysis at Esztergom,
 Hungary, R.~Gibson proved that there are extendable functions
 $f_1,f_2\in\RRR$ such that $f_1+f_2\not\in
\PB$. (See~\cite{PB}.) Thus $\Ext+\Ext\neq\Ext$. Also he asked the
following question.
\begin{Qu}
 If $h\colon\mathR\to\mathR$ is any function, does there exist
 functions $f,g\in\Ext$ such that $f+g=h$?
\end{Qu}
 Answer: {\bf Yes.} H.~Rosen in~\cite{71} proved that an
 arbitrary function $f\colon\mathR\to\mathR$ can be written as
 the sum of two extendable functions. Toward this end he gave an
 example of an extendable function $g\colon\mathR\to\mathR$ whose
 graph is
 dense in $\mathR^2$. In a separate paper~\cite{18},
 K.~Ciesielski and I.~Rec{\l}aw gave the same results.

The problem, whether every bounded function can be written as the
 sum of two bounded functions from a fixed class of Darboux like
 functions has been studied recently in several papers.

\begin{Th}\mbox{}\label{cmal}
\begin{enumerate}
\item[(1)]
 Every bounded function $f\colon\mathR\to\mathR$ is the sum of
 two bounded Darboux functions. {\rm (Maliszewski~\cite{amal})}
\item[(2)]
 Every bounded function $f\colon\mathR\to\mathR$ is the sum of
 two bounded almost continuous functions. {\rm (Ciesielski,
 Maliszewski~\cite{CMal})}
\item[(3)]
 There exists a bounded function $f\colon\mathR\to\mathR$ which
 is not the sum of two bounded functions with perfect road. {\rm
 (Ciesielski, Maliszewski~\cite{CMal})}
\end{enumerate}
\end{Th}
 In particular, Theorem~\ref{cmal} generalizes the result of
 Darji and Humke that every bounded function can be expressed as
 the sum of three bounded almost continuous functions~\cite{24}.
 On the other hand, Theorem~\ref{cmal} shows that the following
 result of Natkaniec is sharp.
\begin{Th}
{\rm (Natkaniec~\cite{64})}
 Every bounded function $f\colon\mathR\to\mathR$ is the sum of
 three bounded extendable functions.
\end{Th}
Note also a surprising result of K.~Ciesielski and J.~Wojciechowski.
\begin{Th}
{\rm (Ciesielski, Wojciechowski~\cite{CW})}
Assume that $n>1$. Then
\begin{enumerate}
\item[(1)]
 Every function $f\colon\mathR^n\to\mathR$ is the sum of $n+1$
 extendable functions.
\item[(2)]
 There exists $f\colon\mathR^n\to\mathR$ that is not the sum of
%ch
 $n$ connectivity functions.
\end{enumerate}
\end{Th}
%ch
Note also that quite recently F.~Jordan constructed a Baire one function
$f\colon\mathR^n\to\mathR$ that is not the sum of $n$ Darboux functions
(unpublished).

\medskip
In 1959, H.~Fast proved the following theorem:
\begin{Th}
{\rm (Fast~\cite{HF})}
 For every family ${\mathcal{F}}\subset\RRR$ with $|{\mathcal
 F}|\leq\co$ there exists $g\in\RRR$ such that $g+f\in\D$ for
 every $f\in {\mathcal F}$.
\end{Th}
 In 1974, K.~Kellum proved the analogous result for the class
 $\ACS$ of almost continuous functions~\cite{49}. On the other
 hand, there is not $g\in\RRR$ such that $g+f\in\D$ for each
 $f\in\RRR$. The problem, for how big families of real functions
 $\mathcal{F}$ there exists such a ``uniform summand'' will be
 study in Subsection~\ref{cardf}. Here we note that such a $g$
 can be found for some regular families ${\mathcal F}\subset\RRR$ of the power
%chTN
$2^{\co}$.
\begin{Th}
 {\rm (Natkaniec~\cite{60})}
 There exists $g\in\RRR$ such that $g+f\in\ACS$ for each
 $f\in\RRR$ with the Baire property (or, for each $f$ that is
 Lebesgue measurable).
\end{Th}
 The similar result was obtained recently for the class $\Ext$ of
 extendable functions.
\begin{Th}\label{now}
%ch
 {\rm (Natkaniec, Rec{\l}aw~\cite{nowe})}
 There exists $g\in\RRR$ such that $g+f\in\Ext$ for each
 $f\in\RRR$ with the Baire property (for each $f$ that is
 Lebesgue measurable).
\end{Th}

%ch
J.~Ceder considered the analogous problem for  classes of Borel measurable
functions.
\begin{Th}\label{ceder-us}
{\rm (Ceder~\cite{jc-us})}
Let $\cal A$ be a countable family of Baire~$\alpha$ functions. Then there
exists a function $f$ such that $f+g$ is a Darboux function of Baire class
$\max(\alpha+1,3)$ for any $g\in{\cal A}$.
\end{Th}
Moreover, Ceder wrote in his paper: {\em ``We do not know, however, whether
or not Theorem~\ref{ceder-us} itself can be valid for families $\cal A$
with cardinality $\co$.''} This problem was solved quite recently by
Rec{\l}aw and Natkaniec.
\begin{Th}
{\rm (Natkaniec, Rec{\l}aw~\cite{nowe})}
For every $\alpha<\omega_1$ there is a Borel measurable function $f$ such
that $f+g\in\ACS$ for any $g\in B_{\alpha}$.
\end{Th}

\medskip
\newcommand{\op}[1]{\operatorname{#1}}
\newcommand{\ma}{{\op{{M}_{a}}}}

For a class $\F\subset\RRR$ we can consider also the {\it maximal
 additive family for~$\F$}:
$$\ma(\F)=\{ g\in\RRR\colon\, f+g\in\F\, \text{ for all $f\in\F$}\}$$
 The maximal additive families for Darboux like functions were
 studied in several papers.
\begin{Th}
\mbox{}
\begin{enumerate}
\item[(1)]
$\ma(\D)={\rm Const}$ {\rm (Radakovi{\v{c}}~\cite{Rad})}.
\item[(2)]
$\ma(\Ext)=\CC$ {\rm (Jastrz{\c{e}}bski, Natkaniec~\cite{JN})}.
\item[(3)]
 $\ma(\ACS)=\ma(\Conn)=\CC$ {\rm (Jastrz{\c{e}}bski,
 J{\c{e}}drzejewski, Natkaniec~\cite{JJN})}.
\item[(4)]
$\ma(\PC)=\ma(\PR)=\CC$ {\rm (K.~Banaszewski~\cite{KB1})}.
\item[(5)]
 If the additivity of category equals $\co$, then
 $\ma(\CIVP)={\rm Const}$ {\rm (K.~Banaszewski~\cite{2})}.
\item[(6)]
 If the additivity of category equals $\co$, then
 $\ma(\SCIVP)={\rm Const}$.
\end{enumerate}
\end{Th}
 The proof of (6) is essentially the same as Banaszewski's proof
 of (5). We are unable to prove those equalities in ZFC.


\subsection{Products}
\begin{Th}{\rm (K.~Banaszewski~\cite{2})}
 Every function $f\colon\mathR\to\mathR$ can be expressed as the
 product of two $\SCIVP$ functions.
\end{Th}

Note also that there are real functions that cannot be written as
 the product of finite many of Darboux functions. The class of
 all products of Darboux functions was characterized by
J.~Ceder.
\begin{Th}
{\rm (Ceder~\cite{JC})}
 A function $f\colon\mathR\to\mathR$ is the product of two
 Darboux functions iff it satisfies the following condition
\begin{description}
\item[(JC)]
$f$ has a zero in each subinterval in which it changes sign.
\end{description}
\end{Th}

The analogous theorem was proved by T.~Natkaniec
for almost continuous functions.
\begin{Th}\label{pro}
{\rm (Natkaniec~\cite{62})}
 Assume that the additivity of category is equal to~$\co$. A real
 function $f\colon\mathR\to\mathR$ is the product of two almost
 continuous functions iff it satisfies the condition (JC).
\end{Th}
 Recently A.~Maliszewski shoved that the extra set-theoretical
 assumption in Theorem~\ref{pro} can be omitted~\cite{Ampro}.
Thus we have the following corollary.
\begin{Co}\label{il}
 Assume that $f\colon\mathR\to\mathR$. The following conditions
 are equivalent
\begin{enumerate}
\item[(i)]
$f$ is the product of two almost continuous functions;
\item[(ii)]
$f$ is the product of two connectivity functions;
\item[(iii)]
$f$ is the product of two Darboux functions;
\item[(iv)]
$f$ possesses the property (JC).
\end{enumerate}
\end{Co}

Corollary \ref{il} suggests the following {\bf open} questions.
\begin{Qu}
 Assume that $f\colon\mathR\to\mathR$ satisfies the condition
 (JC). Is it the product of $\Ext$ ($\ACS\cap\PR$) functions?
\end{Qu}
\begin{Qu}
 Characterize products of $\Ext$ ($\Conn$ or $\D$) functions from
 $\mathR^n$ into $\mathR$.
\end{Qu}

\newcommand{\mm}{{\op{{M}_{m}}}}
\newcommand{\M}{\rm M}

\medskip
 The maximal multiplicative families for Darboux like functions
 were studied in several papers. Recall that
$$\mm(\F)=\{ g\in\RRR\colon\, fg\in\F\, \text{ for all $f\in\F$}\}$$
 Recall also that the maximal multiplicative family for the class
 of all Darboux, Baire one functions is equal to the following
 class $\M$ defined by R.~Fleissner~\cite{RF}.
\begin{description}
\item[M]
 -- $f\in\M$ iff it possesses the following property: if $x_0$ is
 a right-hand (left-hand) point of discontinuity of $f$, then
 $f(x_0)=0$ and there is a sequence $(x_n)$ converging to $x_0$
 such that $x_n>x_0$ ($x_n<x_0$) and $f(x_n)=0$.
\end{description}

\begin{Th}
\mbox{}
\begin{enumerate}
\item[(1)]
$\mm(\D)={\rm Const}$ {\rm (Radakovi{\v{c}}~\cite{Rad})}.
\item[(2)]
$\mm(\Ext)=\M$ {\rm (Jastrz{\c{e}}bski, Natkaniec~\cite{JN})},
\item[(3)]
 $\mm(\ACS)=\mm(\Conn)=\M$ {\rm (Jastrz{\c{e}}bski,
 J{\c{e}}drzejewski, Natkaniec~\cite{JJN})}.
\item[(4)]
$\mm(\PC)=\mm(\PR)=\M$ {\rm (K.~Banaszewski~\cite{KB1})}.
\item[(5)]
 If the additivity of category equals $\co$, then
 $\mm(\CIVP)={\rm Const}$ {\rm (K.~Banaszewski~\cite{2})}.
\item[(6)]
 If the additivity of category equals $\co$, then
 $\mm(\SCIVP)={\rm Const}$.
\end{enumerate}
\end{Th}
 The proof of (6) is essentially the same as Banaszewski's proof
 of (5). We are unable to prove those equalities in ZFC.


\subsection{Maxima and minima}
\begin{Th}
{\rm (Natkaniec~\cite{64})}
Every function
 $f\colon {\mathI}\to {\mathI}$ can be written as
 $\min(\max(f_0,f_1),\max(f_2,f_3))$, where $f_i\in\Ext$ for
 $i=0,1,2,3$.
\end{Th}

Note that the same result can be proved also for real functions
 defined on~$\mathR$ (c.f.,~\cite{18,71}).


Real functions defined on $\mathR$ that are the maximum of
 Darboux functions were characterized by J.~Ceder~\cite{max}.
Such functions that are the maximum of perfect road functions
 were described by K.~Banaszewski~\cite{KB1}.
Thus, notice the following {\bf open} problem.
\begin{Qu}\label{ma}
 Characterize functions that are the maximum of functions from
 other Darboux like classes.
\end{Qu}

A partial solution of Question~\ref{ma} for the class $\ACS$ is
 contained in~\cite{60}.

\subsection{Cardinal functions}\label{cardf}

\newcommand{\add}{{\op{a}}}
\newcommand{\mul}{{\op{m}}}
\newcommand{\cout}{{\op{{c}_{out}}}}
\newcommand{\cin}{{\op{{c}_{in}}}}
\newcommand{\CL}{{\op{{c}_{l}}}}
\newcommand{\CR}{{\op{{c}_{r}}}}


Results from the previous subsections were recently
%ch
strengthened by
 the consideration of the following cardinal functions.
 (This functions were introduced in\cite{60,comp,18}. See also the
survey~\cite{sta}.)
\begin{eqnarray*}
{\add}({\mathcal{F}}) \!\!\!\! &=& \!\!\!\!
\min\left\{\left|{\mathcal{H}}\right|\colon{\mathcal{H}}
\subset\mathR^\mathR\ \&\
\neg\exists g\in\mathR^{\mathR }\
\forall h\in{\mathcal{H}}\ g+h\in{\mathcal{F}}\right\}
\!\cup\!\{(2^\co)^+\}\\
\!\!& = &\!\!\!\!
\min\left\{\left|{\mathcal{H}}\right|\colon{\mathcal{H}}
\subset\RRR\ \&\
\forall g\in\RRR\
\exists h\in{\mathcal{H}}\ g+h\notin{\mathcal{F}}\right\}
\!\cup\!\{(2^\co)^+\}\\
\mbox{}\\
{\mul}({\mathcal{F}}) \!\!\!\! &=& \!\!\!\!
\min\left\{\left|{\mathcal{H}}\right|\colon{\mathcal{H}}
\subset\RRR\ \&\
\neg\exists g\in\RRR\setminus\{ 0\}\
\forall h\in{\mathcal{H}}\ g\cdot h\in{\mathcal{F}}\right\}
\!\cup\!\{(2^\co)^+\}\\
\!\! & = & \!\!\!\!
\min\left\{\left|{\mathcal{H}}\right|\colon{\mathcal{H}}
\subset\RRR\ \&\
\forall g\in\RRR\setminus\{ 0\}\
\exists h\in{\mathcal{H}}\ g\cdot h\notin{\mathcal{F}}\right\}
\!\cup\!\{(2^\co)^+\}.\\
\mbox{}\\
\cout(\mathcal{F})
\!\!\!\!& = & \!\!\!\!
\min\left\{\left|{\mathcal{H}}\right|\!\colon{\mathcal{H}}
\subset\RRR\ \&\
\! \neg{\exists g}\in\RRR\setminus\! \Const\
{\forall h}\in{\mathcal{H}}\ {g\circ h\in{\mathcal{F}}}\right\}
\!\!\cup\!\!\{(2^\co)^+\}\\
\mbox{}\\
\cin({\mathcal{F}}) \!\!\!\! &=& \!\!\!\!
\min\left\{\left|{\mathcal{H}}\right|\!\colon{\mathcal{H}} \subset\RRR\ \&\
\! \neg\exists g\in\RRR\setminus\! \Const\
\forall h\in{\mathcal{H}}\ h\circ g\in{\mathcal{F}}\right\}
\!\!\cup\!\!\{(2^\co)^+\}\\
\mbox{}\\
\CR({\mathcal{F}})
\!\!\!\!& = &\!\!\!\!
\min\left\{\left|{\mathcal{G}}\right|\!\colon{\mathcal{G}}
\subset\RRR\ \&\
\! \neg{\exists h}\in{\mathcal{F}}\
{\forall g}\in{\mathcal{G}}\ {\exists f}\in{\mathcal{F}}\,
 f\circ h=g\right\}
\!\cup\!\{(2^\co)^+\}\\
\mbox{}\\
\CL({\mathcal{F}})
\!\!\!\!& = & \!\!\!\!
\min\left\{\left|{\mathcal{G}}\right|\!\colon{\mathcal{G}}
\subset\RRR\ \&\
\! \neg{\exists h}\in{\mathcal{F}}\
{\forall g}\in{\mathcal{G}}\ {\exists f}\in{\mathcal{F}}\,
 h\circ f=g\right\}
\!\cup\!\{(2^\co)^+\}
\end{eqnarray*}

Values of those cardinal functions for Darboux like classes from
 $\mathR$ to $\mathR$ were investigated by
 T.~Natkaniec~\cite{60,comp,tatra}, K.~Ciesielski and
 A.~W.~Miller~\cite{CM}, K.~Ciesielski and
 I.~Rec{\l}aw~\cite{18}, and T.~Natkaniec and
 I.~Rec{\l}aw~\cite{NR}. Known results are listed in the
 following table.
$$
\begin{array}{|l||c|c|c|c|c|c|c|c|}
 \hline
 \F & \Ext & \ACS & \Conn & \D  & \SCIVP &
\CIVP & \PR  & \PC
\\
 \hline
 {\add}({\F}) & \co^+ & \kappa & \kappa & \kappa & \co^+ & \co^+
 & \co^+ & 2^{\co}\\
 \hline
 {\mul}({\F}) & 2 & \cf(\co) & \cf(\co) & \cf(\co)  &  2 & 2 &  2  & \co \\
 \hline
\cout({\F}) & 1 &  \cf(\co)  & \cf(\co)  & \cf(\co)
 &1 & 1 &1 & \co  \\
 \hline
 \cin({\F}) & 1 & 1 &1 & 1 & 1 & 1 & (2^{\co})^+ & (2^{\co})^+\\
 \hline
 \CR(\F) &  1 &  1 & 1 &  1 & 1 &  1 & (2^{\co})^+ &  (2^{\co})^+ \\
 \hline
\CL(\F) &  1 &  1 & 1 &
 1 & 1 & 1 & (2^{\co})^+ & (2^{\co})^+  \\
 \hline
\end{array}
$$
{\centerline{Table~1.}}

  where $\kappa={\add}(\ACS)={\add}(\Conn)={\add}(\D)$
 satisfies the inequalities $\co<\kappa\leq 2^{\co}$ and
 $\cf(\kappa)>\co$. Moreover, it is consistent with ZFC
 that $\kappa$ can be equal to any regular cardinal between
 $\co^+$ and $2^{\co}$ and that it can be equal to $2^{\co}$
 independently of the cofinality of $2^{\co}$ \cite{CM}.
 Additionally, the equalities $(i,j)$ are proved

\begin{tabular}{llll}
in \cite{18} & for $i=1,2$ & and  & $j=1,7,8$;\\
in \cite{NR} & for $i=2$ & and & $j=2,3,4$;\\
in \cite{comp} & for $i=3,4$ & and & $j=1,2,3,4,5,7,8$;\\
in \cite{tatra} & for $i=5,6$ & and & $j=7,8$.
\end{tabular}

Here $(i,j)$ denotes the coordinates of given equality in our
 table; $i$ denotes the number of the line and $j$ the number of
 the column. The other equalities easily follow from the
 monotonicity of considered cardinal functions.

Table~1 can be easily complete for the other Darboux like classes.
\begin{Th}
 The following equalities hold for the classes $\WCIVP$, $\PB$
 and $\ACS\cap\PR$:
$$
 \begin{array}{|l||c|c|c|}
 \hline
\F & \ACS\cap\PR & \WCIVP & \PB\\
\hline
\add(\F) & \co^+ & \co^+ & \co^+ \\
\hline
\mul(\F) & 2 & 2 & 2\\
\hline
\cout(\F) & 1 & 1 &1 \\
\hline
\cin(\F) & 1 & (2^{\co})^+ & (2^{\co})^+ \\
\hline
\CR(\F) & 1 & (2^{\co})^+ & (2^{\co})^+ \\
\hline
\CL(\F) & 1 & (2^{\co})^+ & (2^{\co})^+ \\
\hline
\end{array}
$$
\center{Table~2.}
\end{Th}

From cardinal functions defined above, the function $\add(\F)$
 has been studied most extensively.
In particular, K.~Ciesielski and J.~Wojciechowski proved
recently the following results.
\begin{Th}\label{cwadd}
{\rm (Ciesielski, Wojciechowski~\cite{CW})}
 In the class of real functions defined on $\mathR^n$, with
 $n>1$, the following equalities hold:
$$ \add(\Ext)=\add(\PC)=\add(\Conn)=\add(\D)=2.$$
\end{Th}

F.~Jordan in~\cite{Jor} considered the function $\add(\F)$ for
complements of Darboux like classes.
 %ch
 An analogous function has been also
 studied for bounded Darboux like functions in~\cite{CMal}.
It is surprising that the result obtained for bounded functions are
essentially different than that described above. In particular,
K.~Ciesielski and A.~Maliszewski proved that
$$\add_b(\ACS)=\add_b(\Conn)=\add_b(\D)=\cf(\co).
$$
D.~Banaszewski studied cardinal functions for additive
Darboux like functions~\cite{DB}.

For other results concerning cardinal functions, see the recent
 survey by K.~Ciesielski~\cite{sta}.


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