%corrected according to the referee remarks by TN, 04/30/98
\documentclass{rae}\usepackage{amsfonts}
\usepackage{amssymb}\usepackage{latexsym}
\usepackage{amsmath}

\newtheorem{tw}{Theorem}[section] 
\newtheorem{df}{Definition}
\newtheorem{lem}{Lemma}[section]
\newtheorem{pro}{Problem}
\newtheorem{cor}{Corollary}[section]
\newcommand{\bd}{\rm bd}
\newcommand\mathN{{\mathbb N}}
\newcommand\mathI{{\mathbb I}}
\newcommand\mathR{{\mathbb R}}
\newcommand\mathQ{{\mathbb Q}}
\newcommand\real{\mathR}
\newcommand{\Qed}{\unskip\nolinebreak\quad\hfill$\Box\;\;$\medskip}

\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{\X}{{\cal X}}
\newcommand{\Ma}{{\cal M}_{a}}
\newcommand{\Mm}{{\cal M}_{m}}
\newcommand{\Ml}{{\cal M}_{l}}
\newcommand{\Const}{{\rm Const}}
\newcommand{\G}{{\cal G}}
\newcommand{\I}{{\cal I}}
\newcommand{\CC}{{\rm C}}
\newcommand{\PC}{{\rm PC}}
\newcommand{\D}{{\rm D}}
\newcommand{\M}{{\rm M}}
\newcommand{\Conn}{{\rm Conn}}
\newcommand{\EE}{{\cal E}}
\newcommand{\Ext}{{\rm Ext}}
\newcommand{\E}{{\rm E}}
\newcommand{\Int}{{\rm int}}
\newcommand{\pf}{{\noindent\sc Proof. }}
\newcommand{\restr}{{\hbox{$|\grave{}$}}}

\author{
Jan Jastrz{\c{e}}bski\footnotemark\addtocounter{footnote}{-1},
Department of Mathematics, Gda{\'n}sk
 University, Wita Stwosza 57, 80-952 Gda{\'n}sk, Poland
(jjas@ksinet.univ.gda.pl) \\ 
 Tomasz Natkaniec\thanks{Both authors are supported by UG
 Research Grant No BW/5100-5-0282-7 and by NSF Cooperative
 Research Grant INT-9600548 with its Polish part financed by
 KBN.}, Department of Mathematics, Gda{\'n}sk University, Wita
 Stwosza 57, 80-952 Gda{\'n}sk, Poland
 (mattn@ksinet.univ.gda.pl)}

\title{On sums and products of extendable functions}
 \date{}

\MathReviews{Primary 26A15; Secondary 54C08.}
 \keywords{extendable functions, peripherally continuous
 functions, family of peripheral intervals, maximal additive
 family, maximal multiplicative family.}
 \markboth{J. Jastrz{\c{e}}bski and T. Natkaniec}{On sums and
 products of extendable functions} 
 
\begin{document}
\maketitle

\begin{abstract}
 We study the maximal additive, multiplicative and lattice-like
 families for the class of all extendable functions.  This
 article is a continuation of earlier papers, in which the same
 questions concerning other Darboux-like functions have been
 studied.
\end{abstract}

\section{Preliminaries}
 Our terminology is standard.  By $\mathR$ and $\mathI$ we denote
 the set of all reals and the interval $[0,1]$, respectively.
 The letters $X$, $Y$ and $Z$ will denote topological spaces. The
 symbols $\overline{A}$, $\Int (A)$ and $\bd (A)$ denote the
 closure, interior and boundary of a set $A$, respectively.

No distinction is made between a function and its graph. For
 functions $f\colon X\to Y$ and $g\colon X\to Z$, the symbol
 $(f,g)$ denotes the diagonal of $f$ and $g$, i.e., $(f,g)\colon
 X\to Y\times Z$, $(f,g)(x)=(f(x),g(x))$ for every $x\in X$.

For a function $f: \mathR \to \mathR$ and $x \in \mathR$ the
 symbols $C^-(f,x)$, $C^+(f,x)$ denote the cluster sets from the
 left and right, respectively, of the function $f$ at the point
 $x$.

By ${\CC}(X,Y)$ and $\Const(X,Y)$ we denote the family of all
 continuous functions and constant functions from $X$ to $Y$,
 respectively. We shall write C and Const when $X$ and $Y$ are
 clear from the context.

We shall consider the following classes of functions from $X$ to $Y$:
\begin{description}
\item[$\PC(X,Y)$]
 --- the class of all peripherally continuous functions.  A
 function $f\colon X\to Y$ 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 $f(\bd (W)) \subset
 V$.
\item[$\E(X,Y)$]
 --- the class of all functions with the property $\EE$.  A
 function $f \colon X \to Y$ has {\it the property $\EE$} if it
 is extendable to a peripherally continuous function, i.e., if
 there exists a peripherally continuous function $F\colon X
\times \mathI \to Y$ such that $f(x)=F(x,0)$ for all $x \in X$.
\item[$\Conn(X,Y)$]
 --- the class of all connectivity functions.  A function
 $f\colon X\to Y$ is {\it connectivity}, if the restriction
 $f\restr C$ is a connected subset of $X\times Y$ whenever $C$ is
 a connected subset of~$X$.
\item[$\Ext(X,Y)$]
 --- the family of {\it extendable\/} functions, i.e.,
 functions $f\colon X\to Y$ for which there exists a connectivity
 function $F\colon X\times \mathI \to Y$ with the property that
 $F(x,0) = f(x)$ for every $x\in X$.
\end{description}

It is well-known that
 $\PC(\mathR^2,\mathR)=\Conn(\mathR^2,\mathR)$. (The inclusion
 ``$\subset$'' was proved by Hamilton \cite{OH} and by Stallings
 \cite{JS}, and the inclusion ``$\supset$'' by Hagan \cite{H}.)
 Therefore, $\E(\mathR,\mathR)=\Ext(\mathR,\mathR)$. 

Moreover, we shall consider the following class of real functions
 of one real variable that was introduced by R.~Fleissner
 \cite{RF}: 
\begin{description}
\item[$\M$]
 -- the class of all functions $f\colon\mathR\to\mathR$ with 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}
 
Recall that if $f\in\M$ then the set $D$ of all points at which
 $f$ is not continuous is nowhere dense, $D\subset f^{-1}(0)$,
 the set $f^{-1}(0)$ is closed, and $f$ is continuous on the
 closure of every component of $\bd(f^{-1}(0))$. Consequently,
 $f$ is a Darboux Baire one function, thus it is also an
 extendable function. (See \cite{BHL}.) 

\medskip
Let ${\X}$ be a class of real functions. The {\it maximal additive 
(multiplicative, lattice-like,} respectively) {\it class} for ${\X}$
is defined to be the class of all $f \in {\X}$ for which $f+g \in {\X}$
$(fg \in {\X}$, $\max(f,g) \in {\X}$ and $\min(f,g) \in {\X}$,
 respectively) whenever 
 $g \in {\X}$. The respective classes are denoted by $\Ma(\X)$,
 $\Mm(\X)$ and $\Ml(\X)$. Those classes for some families of
 Darboux-like functions from $\mathR$ to $\mathR$ were studied by
 several authors. (See, e.g., \cite{Ra} and \cite{Fa} for the
 class of all Darboux functions, \cite{AB} and \cite{RF} for the
 class of all Darboux, Baire one functions, \cite{KB} for the
 class of all perfect road functions and for the class of all
 peripherally continuous functions and \cite{JJN} for the class
 of almost continuous functions, for the class of connectivity
 functions and for the class of functionally connected functions.
 See also the survey \cite{GN} for definitions and relations
 between those properties.) The first systematic study of the
 operators $\Ma(\_)$, $\Mm(\_)$ and $\Ml(\_)$ was done by
 Jastrz\c{e}bski, J\c{e}drzejewski and Natkaniec in \cite{JJN}.
 In particular, they proved the following two basic lemmas. 
\begin{lem}{\rm (\cite[Lemma 2.1]{JJN})}\label{bl1}
 Let $\Phi$ be a property of functions and $X$ be a topological
 space. For $i=1,2$ let $\X_i$ be the class of all functions
 $f\colon X\to\mathR^i$ with the property $\Phi$. Suppose the
 classes $\X_1$ and $\X_2$ fulfill the following conditions: 
\begin{enumerate}
\item[(1.1)]
if $g\in\X_2$ and $h\in\CC(\mathR^2,\mathR)$, then $h\circ g\in\X_1$;
\item[(1.2)]
if $f\in\X_1$ and $g\in\CC(X,\mathR)$, then $(f,g)\in\X_2$.
\end{enumerate}
Then 
$$\CC(X,\mathR)\subset\Ma(\X_1)\cap\Mm(\X_1)\cap\Ml(\X_1).$$
\end{lem}
\begin{lem}{\rm (\cite[Lemma 2.2]{JJN})}\label{bl2}
 Let $\X$ be a family of real functions defined on intervals that
 fulfills the following conditions: 
\begin{enumerate}
\item[(2.0)]
 if $f\in\X$ and $x$ belongs to the domain of $f$, then the sets
 $C^+(f,x)$, $C^-(f,x)$ are connected and $f(x)\in C^+(f,x)\cap
 C^-(f,x)$; 
\item[(2.1)]
 if $f\colon I\to\mathR$, $f\in\X$ and $J$ is a subinterval of an
 interval $I$, then $f\restr J\in\X$; 
\item[(2.2)]
 if $h\colon (a,b)\to\mathR$, $h\in\X$, $y\in C^+(h,a)$, $z\in
 C^-(h,b)$ then the functions $h_1\colon [a,b)\to\mathR$,
 $h_2\colon (a,b]\to\mathR$ and $h_3\colon [a,b]\to\mathR$ belong
 to $\X$, where $h_1=h\cup\{ (a,y)\}$, $h_2=h\cup\{ (b,z)\}$,
 $h_3=h_1\cup h_2$; 
\item[(2.3)]
 if $I\subset\mathR$ is an interval, $a\in I$ and $f\restr(I\cap
 (-\infty,a])\in\X$, $f\restr(I\cap[a,+\infty))\in\X$, then
 $f\in\X$; 
\item[(2.4)]
$\Const\subset\Ma(\X)$ and $-1\in\Mm(\X)$.
\item[(2.5)]
if $f\colon I\to (0,\infty)$ and $f\in\X$, then $1/f\in\X$.
\end{enumerate}
Then
\begin{enumerate}
\item[(i)]
$\Ma(\X)\cup\Ml(\X)\subset\CC$;
\item[(ii)]
$\Mm(\X)\subset\M$.
\end{enumerate}
\end{lem}

We shall employ those lemmas for description of the maximal
 additive, multiplicative and lattice-like classes for the family
 of all extendable functions. In our study we shall use the
 characterization of extendable functions via families of
 peripheral intervals. (See \cite{GR}.) 
 \begin{df} Let $f\colon\mathI \to \mathI $ be a function. A
 family of {\it peripheral intervals} (shortly, PI family) for
 $f$ consists of a sequence of ordered pairs $(I_n,J_n)$ of
 subintervals of $\mathI $ such that 
\begin{enumerate}\label{PI}
 \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}
\end{df}

Gibson and Roush in \cite[Theorems 1 and 2]{GR} proved the
 following theorem: 
\begin{tw} 
 If $f\colon \mathI\to\mathI$ is an extendable function, then
 there exists a PI family for $f$. On the other hand, if 
 for $f\colon\mathI \to \mathI $ there exists a PI family then 
 $f$ is an extendable function. Moreover, then $f$ is the
 restriction of connectivity function $F\colon\mathI^2 \to
 \mathI$ such that $F$ is continuous on the complement of $\mathI
 \times \{ 0 \}$. 
\end{tw}
 It is easy to observe that the analogous characterization is
 valid for any real function defined on an interval. 

\section{Extendable functions}
\begin{lem}\label{l1}
 If $g\in\PC(X,Y)$ and $h\in\CC(Y,Z)$, then $h\circ g
 \in\PC(X,Z)$. 
\end{lem}
\pf
Let $x \in X$, $U$ and $V$ be open neighbourhoods of $x$ 
 and $h(g(x))$, respectively. Then there exists an open
 neighbourhood $W \subset Y$ of the point $g(x)$ such that $h(W)
 \subset V$ and there exists an open neighbourhood $U_0 \subset
 U$ of the point $x$ such that 
$g(\bd U_0) \subset W$. Hence $h(g(\bd U_0)) \subset V$. \Qed

\begin{cor}\label{c1}
 If $g\in\E(X,Y)$ and $h\in\CC(Y,Z)$, then $h\circ g\in\E(X,Z)$.
\end{cor}
\pf
 There exists a peripherally continuous function $G \colon X
 \times I\to Y$ such that $g(x)=G(x,0)$ for all $x \in X.$ Then,
 by Lemma~\ref{l1}, the function $h \circ G \colon X \times I \to
 Z$ is peripherally continuous, and $h \circ g(x)=h \circ G(x,0)$
 for all $x \in X$. So $h \circ g$ has the property ${\cal
 E}$.\Qed 

\begin{lem} \label{l2}
Assume that $X$ is a regular topological space.
 If $f\in\PC(X,Y)$ and $g\in\CC(X,Z)$, then $(f,g) \in\PC(X,Y
 \times Z)$. 
\end{lem}
\pf
Fix $x_0 \in X $. Let $U$ be an open neighbourhood
of $x_0$ and $V$ be an open neighbourhood of $(f(x_0),g(x_0)).$
 There exist open neighbourhoods $V_1 \subset Y$ and $V_2 \subset
 Z$ of $f(x_0)$ and $g(x_0)$, respectively, such that $V_1 \times
 V_2 \subset V$. Let $U_1$ be an open subset of $U$ such that
 $x_0 \in U_1$, $g(U_1) \subset V_2$ and let $U_2 $ be an open
 set such that $\overline{U_2}\subset U_1$, 
$x_0 \in U_2$ and $f(\bd U_2) \subset V_1$. Hence
 $$(f,g)(\bd U_2)\subset f(\bd U_2)\times g(\bd U_2) \subset V_1
 \times V_2 \subset V.$$ 
Thus $(f,g)$ is a peripherally continuous function at $x_0$.\Qed 

\begin{cor} \label{c2}
 Assume that $X$ is a regular topological space. If $f\in\E(X,Y)$
 and $g\in\CC(X,Z)$ then $(f,g)\in\E(X,Y\times Z)$.\Qed 
\end{cor} 

Now, Corollaries \ref{c1}, \ref{c2} and Lemma \ref{bl1} imply the
 following inclusions. 

\begin{cor}\label{c3}
 $${\CC}(\mathR,\mathR)\subset\Ma(\E(\mathR,\mathR))
 \cap\Mm(\E(\mathR ,\mathR ))\cap\Ml(\E(\mathR ,\mathR ))$$ 
\end{cor}

\begin{cor} \label{c4}
$${\CC}(\mathR ,\mathR )\subset\Ma(\Ext(\mathR ,\mathR))
\cap\Mm(\Ext(\mathR,\mathR))\cap\Ml(\Ext(\mathR,\mathR))$$
\end{cor}

Now we shall verify that the class of all extendable real
 functions satisfies all assumptions of Lemma~\ref{bl2}. 
\begin{lem}\label{l21}
Assume that  $g\colon (c, \infty)\to\mathR$ is an extendable 
 function and $y\in C^+(g,c)\cap\mathR$. Then $f=g\cup\{ (c,y)\}$
 is also an extendable function. 
\end{lem}
\pf
 Let ${\cal J}_0$ be a PI family for $g$. For every $n\in\mathN$
 choose $c_n\in (c,c+1]$ such that 
\begin{enumerate}
\item[(a)]  $c_0 =c+1$;
\item[(b)]  $c_n < \min(c+\frac{1}{n}, c_{n-1})$;
\item[(c)]  $|g(c_n)-y|<{\frac{1}{n}}$.
\end{enumerate}
 Put $C=\{ c_n\colon n\in\mathN\}\cup\{ c\}$. Now we shall
 construct a PI family ${\cal J}$ for $f$. A pair $(I,J)$ belongs 
to ${\cal J}$ iff either
\begin{enumerate}
\item[(i)]
$I=[c,c_n)$ and $J=(y-{\frac{1}{n}}, y+{\frac{1}{n}})$ for some 
$n\in\mathN$; or
\item[(ii)]
$(I,J)\in{\cal J}_0$, $I\cap C =\{ c_m\}$ and $g(c_m )\in J$;
 or
\item[(iii)]
$(I,J)\in{\cal J}_0$ and $C\cap I=\emptyset$.
\end{enumerate}

We shall verify that ${\cal J}$ is a PI family for $f$.
 Arrange all elements of ${\cal J}$ in a sequence $(I_n,J_n)_{n
 \in N}$. Then all $I_n$ are open in $[c, \infty )$ and the
 lengths of $I_n$ converge to 0, so the condition (1) from
 Definition~\ref{PI} is satisfied. 

For $x=c$ the condition (2) is clear by (i). For $x \ne c$, (2)
 follows easily from the fact that ${\cal J}_0$ 
is a PI family for $g$. 

The condition (3) is also obvious. Thus we have to verify only
 the condition (4). Fix $n$, $m$ such that $I_n\cap
 I_m\ne\emptyset$ and neither is a subset of the other. Note that
 either $c\not\in I_n$ or $c\not\in I_m$. If $ c \not\in I_n \cup
 I_m$, then $(I_n,J_n), (I_m,J_m)\in {\cal J}_0$ and therefore
 $J_n \cap J_m \ne \emptyset$. So, suppose that 
 $c \in I_n$ and $c \not\in I_m$. Then $I_n=[c,c_{k_n}) $ for
 some $k_n \in N$ and $c_{k_n} \in I_m$. By (ii), $g(c_{k_n}) \in
 J_m$ and, by (i) and (c), $g(c_{k_n}) \in J_n$. Thus $J_n \cap
 J_m \ne \emptyset$.\Qed 

\begin{lem}\label{l22}
 If $c\in\mathR $, $f\colon\mathR\to\mathR$ and $f\restr
 (-\infty, c]$, $f\restr [c, \infty)$ are extendable functions,
 then 
$f$ is also an extendable function.
\end{lem}
\pf
Let ${\cal J}_0$ and ${\cal J}_1$ denote PI families for
$f\restr (- \infty ,c]$ and $f\restr [c, \infty )$ respectively. 
 For every $n\in\mathN$ choose $(I^-_n,J^-_n)\in {\cal J}_0$,
 $(I^+_n,J^+_n)\in {\cal J}_1$ such that $c\in I^-_n\cap I^+_n$
 and the lenght of $I_n^-\cup I_n^+$ is less than $\frac{1}{n}$. 
 Now we shall define a PI family ${\cal J}$ for $f$. A pair
 $(I,J)$ belongs to ${\cal J}$ iff either 
\begin{enumerate}
\item[(i)]
$(I,J)\in{\cal J}_0\cup{\cal J}_1$ and $c\not\in I$; or
\item[(ii)]
$I=I_n^- \cup I_n^+$ and $J=J_n^- \cup J_n^+$ for some $n\in \mathN$.
 \end{enumerate}
It is easy to verify that  ${\cal J}$ is a PI family for $f$. \Qed

From Lemmas \ref{bl2}, \ref{l21} and \ref{l22} we obtain the
 following inclusions: 
\begin{cor}\label{c5} 
\begin{eqnarray*}
 \Ma(\Ext(\mathR ,\mathR ))\cup\Ml(\Ext(\mathR,\mathR)) & \subset
 & \CC(\mathR , \mathR)\\ 
\Mm(\Ext(\mathR ,\mathR ))  & \subset & \M
\end{eqnarray*}
\end{cor}

Thus, Corollaries \ref{c4} and \ref{c5} yield the following equalities:
\begin{tw} 
 $$\Ma(\Ext(\mathR,\mathR))=\CC(\mathR ,\mathR)= \Ml(\Ext(\mathR
 ,\mathR))$$ 
\end{tw}

\begin{lem}\label{lp}
Assume that $C$ is a nowhere dense closed subset of $\mathR$
and $g\colon\mathR\to\mathR$ satisfies the following conditions:
\begin{enumerate}
\item[(a)]
$g(x)=0$ for $x\in C$;
\item[(b)]
if $J\subset\mathR$ is a component of $\mathR \setminus
C$, then $g\restr\overline{J}$ is an extendable function.
\end{enumerate}
Then $g$ is an extendable function.
\end{lem}
\pf
Let $\{ (a_n,b_n)\}_n$  be the sequence of all components of 
 $\mathR \setminus C$. For each $n$ let ${\cal K}_n$ be a PI
 family for $g\restr\overline{(a_n,b_n)}$. 
 For every positive integer $n$ choose a finite family $\I_n$ of
 open intervals such that 
\begin{itemize}
\item
$C\subset\bigcup\I_n$;
\item 
the length of each $I\in\I_n$ is less than $1/n$;
\item
the end-points of every $I\in\I_n$ belong to $I\setminus C$;
\item
 if $I\in\I_n$, $\inf(I)\in (a_i,b_i)$, $\sup(I)\in (a_j,b_j)$,
 $I^-=I\cap (a_i,b_i]$ and $I^+=I\cap [a_j,b_j)$, then there are
 intervals $J^-$, $J^+$ such that $(I^-,J^-)\in\K_i$,
 $(I^+,J^+)\in\K_j$ and the length of $J^-\cup J^+$ is less than
 $1/n$. 
\end{itemize}
 We shall define a PI family ${\cal J}$ for $g$. A pair $(I,J)$
 belongs to ${\cal J}$ iff either 
\begin{enumerate}
\item[(i)]
there exists $n$ such that $(I,J) \in {\cal K}_n$ and $I \cap C=
\emptyset $; or
\item[(ii)]
 $I\in\I_n$ for some $n$ and $J=J^-\cup J^+$, where $J^-$, $J^+$
 are described above. 
\end{enumerate}

Now we shall verify that ${\cal J}$ is a PI family for $g$. We
 can arrange all elements of ${\cal J}$ in a sequence $\{
 (I_n,J_n)\}_n$ such that the lengths of $I_n$ 
 converge to 0, so the condition (1) from Definition \ref{PI} is
 satisfied. 

To prove the condition (2), fix $x \in \mathR$ and
 $\varepsilon>0$. If $x\not\in C$, then $x\in (a_m, b_m)$ for
 some $m$. There exists $(I,J) \in {\cal K}_m$ that fulfills~(2)
 and, by~(i), $(I,J)\in{\cal J}$. If $x \in C$, then there exist
 $n\in \mathN$ and $(I,J)\in{\cal J}$ such that $I\in\I_n$,
 $1/n<\varepsilon$ and $x\in I$. Then $g(x)=0\in J$ and the
 lengths of $I$ and $J$ are less than $1/n$, so the pair $(I,J)$
 satisfies~(2). 

The statement (3) is obvious by the definition of~${\cal J}$.

To verify (4), fix $(I_n, J_n)$ and $(I_m, J_m)$ such that $I_n
 \cap I_m\ne \emptyset$ and neither is a subset of the other.
 Note that $0 \in J_k$ whenever $I_k\cap C \ne \emptyset$. Thus,
 if $I_n\cap C\ne\emptyset\ne I_m\cap C$, then $0 \in J_n \cap
 J_m$. 

If $(I_n \cup I_m) \cap C = \emptyset$, then $I_n \cup I_m
 \subset (a_i, b_i)$ for some $i \in \mathN$ and therefore
 $(I_n,J_n), (I_m,J_m) \in {\cal K}_i$. Thus $J_n \cap J_m \ne
 \emptyset $. 

So, assume that $I_n\cap C\ne\emptyset$ and $I_m\cap
 C=\emptyset$. Then $I_m \subset (a_i, b_i)$ for some $i \in
 \mathN$ and either $a_i\in I_n$ or $b_i\in I_n$. Suppose that
 $b_i \in I_n$. By the definition of ${\cal J}$ there exist 
 $(I_0, J_0) \in {\cal K}_i$ such that $I_0 = I_n\cap (a_i, b_i]$
 and $J_0\subset J_n$. Because $(I_m,J_m) , 
(I_0, J_0) \in {\cal K}_i$ and $I_m \cap I_0 \ne \emptyset$, then
 $J_m \cap J_0 \ne \emptyset$. Consequently, $J_m\cap
 J_n\neq\emptyset$, so ${\cal J}$ satisfies the
 condition~(4).\Qed 

\begin{tw} 
$$\Mm(\Ext(\mathR ,\mathR))= \M.$$
\end{tw}
\pf
By Corollary \ref{c5}, it is enough to prove that 
$$\M\subset\Mm(\Ext(\mathR ,\mathR)).$$
 Assume that $f\in\Ext(\mathR,\mathR)$ and $g\in\M$. Then
 $D=\bd(f^{-1}(0))$ is a closed and nowhere dense set. Let $J$ be
 a component of the complement of $D$. Then $g$ is continuous on
 $\overline{J}$ and, by 
 Corollary~\ref{c4}, $fg$ is extendable on $\overline{J}$.
 Moreover, $fg(x)=0$ for $x \in D$, and according to
 Lemma~\ref{lp}, $fg$ is an extendable function.\Qed 

\section{Applications}
 The next lemma shows that in the definition of extendability
 (for real functions) we can replace the compact interval
 $\mathI$ by whole real line. 
\begin{lem}
 For a function $f \colon\mathR\to\mathR$ the following
 conditions are equivalent: 
\begin{enumerate}
\item[(i)]
$f\in\Ext(\mathR ,\mathR)$;
\item[(ii)]
 there is $F\in\PC(\mathR^2,\mathR)$ such that $f(x)=F(x,0)$ for
 each $x\in\mathR$. 
\end{enumerate}
\end{lem}
\pf
 Let $f\in\Ext(\mathR ,\mathR)$. Then there exists
 $F_0\in\PC(\mathR\times\mathI,\mathR)$) such that
 $f(x)=F_0(x,0)$ for $x\in\mathR $. According to \cite[Theorem
 2]{GR}, we can assume that $F_0$ is continuous on $\mathR \times
 (0,1]$. Thus $F_+\colon\mathR\times [0,\infty) \to\mathR$
 defined by 
$$F_+(x,y)= \left\{
\begin{array}{lll}
F_0(x,y) &\mbox{ for }& (x,y)\in\mathR\times\mathI \\
F_0(x,1) & \mbox{ for }& (x,y)\in\mathR\times [1, \infty)
\end{array}
\right.$$
 is a peripherally continuous function and consequently
 $F\colon\mathR ^2 \to 
\mathR $ defined by
$$F(x,y)=\left\{
\begin{array}{lll}
F_+(x,y) &\mbox{ for }& (x,y)\in\mathR\times [0, \infty)\\
F_+(x,-y) &\mbox{ for }& (x,y)\in\mathR\times (-\infty ,0]
\end{array}
\right.$$
 is also a peripherally continuous function with $f(x)=F(x,0)$
 for every $x\in\mathR$. 

Now, if $F\in\PC(\mathR^2,\mathR)$ and $f(x)=F(x,0)$ for each $x
 \in \mathR $, then $F\restr(\mathR\times\mathI)$ is also a
 peripherally continuous extension of $f$. Thus
 $f\in\Ext(\mathR,\mathR)$. \Qed 

\begin{tw}
$$\Ma(\PC(({\mathR}^2,\mathR)) =\CC({\mathR}^2, \mathR)
=\Ml(\PC({\mathR}^2,\mathR))$$
\end{tw}
\pf
By Lemmas \ref{l1} and \ref{l2}, 
$$\CC({\mathR}^2,\mathR)\subset\Ma(\PC({\mathR}^2, \mathR)).$$
On the other hand,
$$\Ma(\PC({\mathR}^2,\mathR))\subset\PC({\mathR}^2, \mathR)$$
because $f\equiv 0$ belongs to $\PC({\mathR}^2,\mathR)$. Suppose that
$g\in\PC({\mathR}^2, \mathR)$ is discontinuous at $x_0 \in {\mathR}^2.$
 Let $h\colon\mathR\to{\mathR}^2$ be a homeomorphic injection of
 $\mathR$ into ${\mathR}^2$ such that 
\begin{itemize}
\item
$h(0)= x_0$ and $g \circ h$ is discontinuous at 0;
\item
 there is a homeomorphism $h_1\colon{\mathR}^2 \to {\mathR}^2$
 such that $h_1(x,0)=h(x)$ for $x\in\mathR$. 
\end{itemize}
 By Corollary \ref{c5}, there is $f_0\in\Ext(\mathR,\mathR)$ such
 that $f_0+g\circ h\not\in\Ext( \mathR,\mathR ).$ Let
 $f_1\colon\mathR^2\to\mathR$ be a peripherally continuous 
extension of $f_0$, i.e., $f_1(x,0)=f_0(x)$ for $x\in\mathR$. Then
$f=f_1 \circ h_1^{-1}\in\PC({\mathR}^2,\mathR)$.
Suppose that $f+g \in\PC({\mathR}^2,\mathR)$. Then
$$f_1+g \circ h_1 = (f+g) \circ h_1 \in\PC({\mathR}^2, \mathR).$$
On the other hand, for each $x \in \mathR$ we have
$$(f_1+g \circ h_1)(x,0)=f_1(x,0)+g(h_1(x,0))=f_0(x)+g \circ h(x).$$
 Thus $f_0+g \circ h \in\Ext(\mathR ,\mathR)$, contrary to the
 choice of $f_0$. Consequently, 
$$\Ma(\PC({\mathR}^2, \mathR)) \subset \CC({\mathR}^2, 
\mathR).$$ 

In a similar way we can prove that 
$$\Ml(\PC({\mathR}^2,\mathR )) =\CC({\mathR}^2,\mathR).$$ 
To see it, note that 
$$\Ml(\PC({\mathR}^2,\mathR))\subset\PC({\mathR}^2, \mathR).$$
 Indeed, suppose that $f\colon \mathR^2\to\mathR$ is not
 peripherally continuous at $x_0\in\mathR^2$. Then there are an
 open neighbourhood $W$ of $x_0$ and $\varepsilon>0$ such that
 $f(\bd(U))\subset (f(x_0)-\varepsilon, f(x_0)+\varepsilon)$ for
 no open neighbourhood $U$ of $x_0$ with $U\subset W$. Then the
 function $\max(f, f(x_0)-\varepsilon)$ is not peripherally
 continuous at $x_0$ and the constant function
 $f(x_0)-\varepsilon$ is peripherally continuous. Thus
 $f\not\in\Ml(\PC({\mathR}^2,\mathR))$. 
\Qed

Finally, note that Lemmas \ref{l1} and \ref{l2} yield the
 following inclusion 
$$\CC(\mathR^2,\mathR)\subset\Mm(\PC(\mathR^2,\mathR)).$$
 On the other hand, it is easy to verify that
 $f\colon\mathR^2\to\mathR$ defined by 
$$f(x)=\left\{
\begin{array}{cl}
\sin(||x||^{-1}) & \mbox{ if $x\neq 0$;}\\
0 & \mbox{ if $x=0$}
\end{array}
\right.$$
 is a discontinuous function that belongs to
 $\Mm(\PC(\mathR^2,\mathR))$. Thus we finish with the following
 problem: 
\begin{pro}
Characterize the class $\Mm(\PC(\mathR^2,\mathR))$.
\end{pro}

\medskip

\begin{thebibliography}{21}
\bibitem[KB]{KB}
 K.~Banaszewski,  {\em Algebraic properties of
${\cal E}$-continuous functions}, Real Anal. Exchange
{\bf 18} (1992--93), 153--168.

\bibitem[BHL]{BHL}
 J.~Brown, P.~Humke and M.~Laczkovich, {\em Measurable Darboux
 functions}, Proc. Amer. Math. Soc. {\bf 102} (1988), 603--610. 

\bibitem[AB]{AB}
 A.~M.~Bruckner, {\em Differentiation of Real Functions}, Lecture
 Notes in Mathematics, Vol. {\bf 659} Springer-Verlag, Berlin,
 Heidelberg, New York, 1978. 

\bibitem[Fa]{Fa}
 J.~Farkov{\'a}, {\em About the maximum and the minimum of
 Darboux functions}, Mat. {\v{C}}as. Slov. Akad. Vied {\bf 21}
 (1971), 110--116. 

 \bibitem[RF]{RF}
 R.~J.~Fleissner, {\em A note on Baire 1 Darboux functions}, Real
 Anal. Exchange {\bf 3} (1977--78), 104--106. 

\bibitem[GN]{GN}
R.~G.~Gibson and T.~Natkaniec, {\em Darboux like functions}, 
Real Anal. Exchange {\bf 22} (1996--97), 492--534.

\bibitem[GR]{GR} 
 R.~G.~Gibson and F.~Roush, {\em A characterization of extendable
 connectivity functions}, Real Anal. Exchange {\bf 13}
 (1987--88), 214--222. 

\bibitem[H]{H}
 M.~R.~Hagan, {\em Equivalence of connectivity maps and
 peripherally continuous transformations}, Proc. Amer. Math. Soc.
 {\bf 17} (1966), 175--177. 

\bibitem[OH]{OH}
 O.~H.~Hamilton, {\em Fixed points for certain noncontinuous
 transformations}, Proc. Amer. Math. Soc. {\bf 8} (1957), 750--756.

\bibitem[JJN]{JJN}
 J.~Jastrz{\c{e}}bski,~ J.~J{\c{e}}drzejewski and T.~Natkaniec,
 {\em On some subclasses of Darboux functions}, Fund. Math. {\bf
 138} (1991), 165--173. 

\bibitem[Ra]{Ra}
 T.~Radakovi\'{c}, {\em \"{U}ber Darbouxsche und stetige
Funktionen}, Monat. Math. Phys. {\bf 38} (1931), 111--122. 

\bibitem[JS]{JS}
 J.~Stallings, {\em Fixed point theorems for connectivity maps},
 Fund. Math. {\bf 47} (1959), 249--263. 

\end{thebibliography}



\end{document}

