%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}