\documentstyle[11pt]{article} %\documentstyle[11pt]{amsart} \setlength{\topmargin}{.1in} \setlength{\textheight}{8in} \setlength{\textwidth}{5.8in} \setlength{\evensidemargin}{.4in} \setlength{\oddsidemargin}{.4in} \def\mathR{\mbox{$\bf \rm I\makebox [1.2ex] [r] {R}$}} \newcommand{\subRe}{\mbox{\scriptsize $\bf \rm I\makebox [1.2ex] [r] {R}$}} \newcommand{\Bbb}[1]{I\!\!#1} \newtheorem{Th}{Theorem} \newtheorem{Pro}{Problem} \newtheorem{Ex}{Example} \newtheorem{Le}{Lemma} \newtheorem{Co}{Corollary} \newtheorem{Pn}{Proposition} \renewcommand{\labelenumi}{(\arabic{enumi})} \renewcommand{\theenumi}{(\arabic{enumi})} \newcommand{\pf}{{\noindent\sc Proof. }} \newcommand{\Qed}{\nopagebreak\mbox{}\hfill$\Box\;\;$\medskip} %\newcommand{\mathN}{{\Bbb N}} \newcommand{\mathN}{{\bf N}} \newcommand{\mathI}{\/\mbox{\/\rm \bf I}} \newcommand{\mathQ}{{\Bbb Q}} %\def\goth{\frak} %\def\continuum{{\goth c}} %\def\co{{\continuum}} \newcommand{\co}{2^{\omega}} %\newcommand{\mathR}{{\Bbb R}} %\newcommand{\real}{{\Bbb R}} \newcommand{\real}{{\mathR}} %\newcommand{\mathZ}{{\Bbb Z}} \newcommand{\mathZ}{{\bf Z}} %\newcommand{\poset}{{\Bbb P}} \newcommand{\poset}{{\bf P}} \newcommand{\card}{\mbox{\/\rm card}\,} \newcommand{\add}{\mbox{\/\rm add}\,} \newcommand{\cov}{\mbox{\/\rm cov}\,} \newcommand{\unif}{\mbox{\/\rm non}\,} \newcommand{\lin}{\mbox{\/\rm lin}\,} \newcommand{\dom}{\mbox{\/\rm dom}\,} \newcommand{\rng}{\mbox{\/\rm rng}\,} \newcommand{\INT}{\mbox{\/\rm int}\,} \def\la{\langle} \def\ra{\rangle} %\author{Marek Balcerzak} %\author{Krzysztof Ciesielski} %\author{Tomasz Natkaniec} %\title{On a problem of Darji} %\author{} %\title{} \begin{document} %\maketitle %\ch I like to have address with the names. %\chTN { \renewcommand{\thefootnote}{} \footnotetext{AMS Subject Classification (1991): Primary: 26A15;} } { {\renewcommand{\thefootnote}{} \footnotetext{Key words: Sierpi{\'n}ski-Zygmund functions, almost continuous functions, extendable functions, the Cantor intermediate value property CIVP, the stong Cantor intermediate value property SCIVP.} } \null\bigskip \begin{center} \Large\bf Sierpi\'nski--Zygmund functions that have the Cantor intermediate value property \end{center} \bigskip Krzysztof Banaszewski, Department of Mathematics, Pedagogical University, Chodkiewi\-cza~30, 85-064 Bydgoszcz, Poland \\[\medskipamount] Tomasz Natkaniec, Department of Mathematics, Pedagogical University, Chodkiewi\-cza~30, 85-064 Bydgoszcz, Poland (wspb11@cc.uni.torun.pl) \vskip-.5in \begin{abstract} We construct (in ZFC) an example of Sierpi\'nski--Zygmund function having the Cantor intermediate value property and observe that every such function does not have the strong Cantor intermediate value property, which solves the problem of R.~Gibson \cite[Question~2]{RG}. Moreover we prove that both families: $SCIVP$ functions and $CIVP\setminus SCIVP$ functions are $2^{\co}$ dense in the uniform closure of the class of $CIVP$ functions. We show also that if the real line $\mathR$ is not a union of less than continuum many its meager subsets, then there exists an almost continuous Sierpi\'nski--Zygmund function having the Cantor intermediate value property. Because such a function does not have the strong Cantor intermediate value property, it is not extendable. This solves another problem of Gibson \cite[Question 3]{RG}. \end{abstract} \section{Introduction} Our terminology is standard. We shall consider only real-valued functions of one real variable. No distinction is made between a function and its graph. The family of all functions from a set $X$ into $Y$ will be denoted by $Y^X$. Symbol $\card(X)$ will stand for the cardinality of a set $X$. The cardinality of $\mathR$ is denoted by $\co$. If $A$ is a planar set, we denote its $x$-projection by $\dom(A)$. For $f,g\in\mathR^{\subRe}$ the notation $[f=g]$ means the set $\{x\in\mathR\colon f(x)=g(x)\}$. Recall also the following definitions. \begin{itemize} \item $f\colon \mathR\to\mathR$ is of {\it Sierpi\'{n}ski-Zygmund type\/} (shortly, $f\in SZ$, or $f$ is of S-Z type) if its restriction $f|M$ is discontinuous for each set $M\subset\mathR$ with $\card (M)=\co$~\cite{SZ}. \item $f\colon \mathR\to\mathR$ {\it has a perfect road\/} at $x\in\mathR$ if there exists a perfect set $C$ such that $x$ is a bilaterally limit point of $C$ and $f|C$ is continuous at $x$. We say that $f$ is of {\it perfect road type\/} (shortly, $f\in PR$) if $f$ has a perfect road at each point~\cite{Max}. \item $f\colon\mathR\to\mathR$ is said to be {\it almost continuous\/} in the sense of Stallings (shortly, $f\in AC$) if each open subset of the plane containing $f$ contains also a continuous function $g\colon \mathR\to\mathR$~\cite{St}. \item $F\colon\mathR\times [0,1]\to\mathR$ is {\it connectivity} if the graph of every its restriction $F|X$ is connected in $\mathR^3$ for every connected $X\subset\mathR\times [0,1]$. \item $f\colon\mathR\to\mathR$ is {\it extendable} (shortly, $f\in Ext$) if there is a connectivity function $F\colon \mathR\times [0,1]\to \mathR$ such that $F(x,0) = f(x)$ for every $x\in\mathR$. \item $f\colon\mathR\to\mathR$ has the {\it Cantor intermediate value property\/} (CIVP), if for every $x,y\in \mathR$ and for each Cantor set $K$ between $f(x)$ and $f(y)$ there is a Cantor set $C$ between $x$ and $y$ such that $f(C)\subset K$ \cite{CIVP}. \item $f\colon\mathR\to\mathR$ has the {\it strong Cantor intermediate value property\/} (SCIVP), if for every $x,y\in \mathR$ and for each Cantor set $K$ between $f(x)$ and $f(y)$ there is a Cantor set $C$ between $x$ and $y$ such that $f(C)\subset K$ and $f|C$ is continuous \cite{RGR}. \item $f\colon\mathR\to\mathR$ has the {\it weak Cantor intermediate value property\/} (WCIVP), if for every $x,y\in \mathR$ with $f(x)\neq f(y)$ there exists a Cantor set $C$ between $x$ and $y$ such that $f(C)$ is between $f(x)$ and $f(y)$ \cite{WCIVP}. \item $f\colon\mathR\to\mathR$ is a {\it peripherally continuous\/} (shortly, $f\in PC$) if for every $x\in\mathR$ there are sequences $a_{n}\nearrow x$ and $b_{n}\searrow x$ such that $\lim_{n\to\infty} f(a_{n})=\lim_{n\to\infty} f(b_{n})=f(x)$. \end{itemize} The relationships between those classes were discussed in many papers. (See e.g., \cite{RG}.) In particular, the following implications hold. \begin{picture}(0,95) \put(80,50){\makebox(0,0){$Ext$}} \put(90,52){\vector(1,1){20}} \put(90,48){\vector(1,-1){20}} \put(130,72){\makebox(0,0){$AC$}} \put(135,30){\makebox(0,0){$SCIVP$}} \put(145,72){\vector(1,0){105}} \put(260,72){\makebox(0,0){$D$}} \put(160,30){\vector(1,0){20}} \put(205,30){\makebox(0,0){$CIVP$}} \put(225,30){\vector(1,0){20}} \put(260,30){\makebox(0,0){$PR$}} \put(275,30){\vector(1,1){20}} \put(275,72){\vector(1,-1){20}} \put(305,50){\makebox(0,0){$PC$}} \end{picture} At Banach Center in Warsaw, in 1989, Jerry Gibson gave a talk in which he posed several problems, in particular: \begin{Pro} {\rm \cite[Question~2]{RG}} Does $CIVP\Longrightarrow SCIVP$? \end{Pro} and \begin{Pro} {\rm \cite[Question~3]{RG}} Does $AC+CIVP\Longrightarrow Ext$? \end{Pro} In this paper we solve both those problems in the negative. To see it, we shall construct an example of an S-Z function with the Cantor intermediate value property. This generalize the rezult of Darji from \cite{Da}. Because no S-Z function has the strong Cantor intermediate value property, $CIVP\setminus SCIVP\neq \emptyset$. Moreover we generalize Theorem~1 from \cite{BCN} by showing that under the assumption that the real line is not the union of less than continuum many meager sets (which is somewhat weaker than CH or the Martin's Axiom~MA~\cite{Sh}) there exists an almost continuous S-Z function having the CIVP. (On the other hand, there is a model of ZFC in which there is no Darboux S-Z function \cite{BCN}. Thus, the additional set theoretical assumptions are necessary in the result mentioned above.) Again, since $SZ\cap SCIVP=\emptyset$ and $Ext\subset SCIVP$ \cite{RGR}, we obtain $AC+CIVP\neq Ext$. \section{S-Z functions having the CIVP}\label{sec2} In our constructions we will use the following easy and well-known lemmas. \begin{Le}\label{lemGDelta} {\rm \cite{SZ,KK}} Suppose $U\subset\mathR$ and $f\colon U\to\mathR$ is continuous. Then there exists a $G_{\delta}$ set $M$ containing $U$ and a continuous function $g\colon M\to\mathR$ such that $g|U=f$.\Qed \end{Le} \begin{Le}\label{rozl} {\rm \cite[Lemma 3.1]{KB}} Let $J=(a,b)$, $f\in {\cal U}_{0}\cap WCIVP$, $A=f^{-1}(J)$ and denote by $\{I_{m}\}_{m=1}^{\infty}$ the set of all intervals having rational endpoints for which $I_{m}\cap A\neq \emptyset$. If $A\neq \emptyset $ then there exists a sequence of pairwise disjoint Cantor sets $\{K_{m}\}_{m=1}^{\infty}$ such that $K_{m}\subset A\cap I_{m}$ for $m\in \Bbb{N}$. \end{Le} \begin{Th} There exists a Sierpi{\'n}ski--Zygmund function having the CIVP. \end{Th} \pf Let $\{ x_{\alpha}\colon \alpha <\co\}$ be a one-to-one enumeration of $\mathR$, $\{I_n\colon n<\omega\}$ be a sequence of all open intervals with rational end-points, $\{ g_{\alpha}\colon{\alpha <\co}\}$ be an enumeration of all continuous functions defined on $G_{\delta}$ subsets of~$\mathR$ and $\{ C_{\alpha}\colon\alpha<\co\}$ be an enumeration of all Cantor sets (i.e., non-empty compact perfect nowhere dense subsets of the line). It is well-known (and easy to prove) that there exists a family $\{ K_{n,\alpha}\colon\ n<\omega,\ \alpha<\co\}$ of pairwise disjoint Cantor sets such that $K_{n,\alpha}\subset I_n$ for each $n<\omega$ and $\alpha<\co$. (See, e.g., \cite{KB}.) Now, define the values $f(x_\alpha)$ of function $f$ by induction on $\alpha<\co$ as follows. \begin{description} \item[(a)] $f(x_\alpha)\in C_{\beta}\setminus\{ g_{\gamma}(x_{\alpha})\colon \gamma\leq\alpha\}$ provided $x_\alpha\in \bigcup_{n<\omega}K_{n,\beta}$. \item[(b)] $f(x_\alpha)\in\mathR \setminus \{g_\gamma(x_\alpha)\colon \gamma\leq\alpha\}$ otherwise. \end{description} We will show that $f$ has the desired properties. To prove that $f\in CIVP$ fix $x,y\in\mathR$ and a Cantor set $C$ between $f(x)$ and $f(y)$. There exist $n<\omega$ and $\beta<\co$ such that $I_n\subset (x,y)$ and $C=C_{\beta}$. Then $K_{n,\beta} \subset I_n$ and, by (a), $f(K_{n,\beta})\subset C_{\beta}$. Thus $f$ has the CIVP. To prove that $f\in SZ$, by Lemma~\ref{lemGDelta} it is enough to show that $\card([f=g_{\beta}])<\co$ for each $\beta <\co$. But $[f=g_{\beta}]\subset\{x_\alpha\colon\alpha<\beta\})$, so $\card([f=g_{\beta}])<\co$. Hence, $f\in SZ$. \Qed Since $SZ\cap SCIVP=\emptyset$, we obtain the following \begin{Co} $CIVP\neq SCIVP$. \end{Co} Moreover, we shall prove in the next theorem that both sets $CIVP\setminus SCIVP$ and $SCIVP$ are dense in the uniform closure of the class $CIVP$. Recall that this closure is equal to the class ${\cal U}\cap WCIVP$~\cite{KB}. Here $\cal U$ denote the uniform limit of the class of Darboux functions, i.e., the class of all functions $f$ such that for every interval $J$ and every set $A$ of cardinality less than $\co$, the set $f(J\setminus A)$ is dense in the interval $[\inf_{J}f,\sup_{J}f]$ \cite{BCW}. (Note also that ${\cal U}\cap WCIVP={\cal U}\cap PR$.) \begin{Th} For every $\varepsilon>0$ and each $h\in{\cal U}\cap WCIVP$ the sets $\{f\in SCIVP\colon ||h-f||<\varepsilon\}$ and $\{k\in CIVP\setminus SCIVP\colon ||h-k||<\varepsilon\}$ have cardinality equal to $2^{\co}$. \end{Th} \pf Let $\{ x_{\alpha}\colon \alpha <\co\}$ be a one-to-one enumeration of $\mathR$ and $\{g_{\alpha}\colon{\alpha <\co}\}$ be an enumeration of all continuous functions defined on $G_{\delta}$ subsets of~$\mathR$. Choose a sequence $\{ J_n\}_n$ of half open intervals, each of length $\varepsilon$, such that $\rng(h)\subset \INT\bigcup J_n$ and $\INT J_n\cap \rng(h)\neq\emptyset$. For every $n$ let $\{r_{n,\beta}\}_{\beta <\co}$ be a net of all points of $J_n$ and $\{ C_{n,\alpha}\colon\alpha<\co\}$ be an enumeration of all Cantor sets contained in $J_n$. Set $A_n=h^{-1}(\INT J_n)$. Let $\{ I_{n,m}\}_m$ be a sequence of all open intervals with rational end-points such that $I_{n,m}\cap A_n\neq \emptyset$. By Lemma~\ref{rozl}, for every $n$ there exists a sequence $\{ K_{n,m}\}_m$ of pairwise disjoint Cantor sets such that $K_{n,m}\subset I_{n,m}\cap A_n$. Moreover, we can require that $\card(A_n\setminus\bigcup_m K_{n,m})=\co$. Decompose each $K_{n,m}$ into $\co$ many Cantor sets $\{ K_{n,m,\alpha}\}_{\alpha <\co}$. Let $\cal F$ and $\cal K$ be families of functions such that \begin{enumerate} \item[(a)] $f(x_\alpha)=r_{n,\beta}$ and $k(x_\alpha)\in C_{n,\beta}\setminus \{ g_{\gamma}(x_{\alpha})\colon\gamma\leq\alpha\}$ provided $f\in{\cal F}$, $k\in{\cal K}$ and $x_\alpha\in \bigcup_{m=1}^{\infty}K_{n,m,\beta}$. \item[(b)] $f(x_{\alpha })\in J_n$ and $k(x_\alpha)\in J_n\setminus \{g_\gamma(x_\alpha)\colon \gamma\leq\alpha\}$ if $f\in {\cal F}$, $k\in{\cal K}$ and $x_{\alpha}\in h^{-1}(J_n)\setminus\bigcup_{m=1}^{\infty}K_{n,m}$. \end{enumerate} Note that for every $x\in\bigcup_{n,m}K_{n,m}$ the value $k(x)$ can be any element from the set of size $\co$, so $\card({\cal K})=2^{\co}$. Similarly, for every $x\in h^{-1}(J_n)\setminus\bigcup_mK_{n,m}$, $f(x)$ can be any element from the interval $J_n$, so $\card({\cal F})=2^{\co}$. Observe that for each $x\in\mathR$, $f\in{\cal F}$ and $k\in{\cal K}$, if $h(x)\in J_n$ then $f(x),\,k(x)\in J_n$. Therefore $||f-h||<\varepsilon$ and $||k-h||<\varepsilon$. To prove that ${\cal F}\subset SCIVP$ and ${\cal K}\subset CIVP$ fix $f\in{\cal F}$, $k\in{\cal K}$, $x,y\in\mathR$, and Cantor sets $C^{1}$ between $f(x)$ and $f(y)$ and $C^{2}$ between $k(x)$ and $k(y)$, respectively. We can assume that $C^{1}\subset J_{n_{1}}$ and $C^{2}\subset J_{n_{2}}$ for some $n_1,\,n_2$. Let $r\in C^{1}$. Then there exist $\beta_{1},\,\beta_{2} <\co$ such that $r=r_{n_{1},\beta_{1}}$, $C^{2}=C_{n_{2},\beta_{2}}$ and $m_{1},\,m_{2}<\omega$ such that $I_{n_{1},m_{1}}\cup I_{n_{2},m_{2}}\subset (x,y)$ and $I_{n_{1},m_{1}}\cap A_{n_{1}}\neq\emptyset\neq I_{n_{2},m_{2}}\cap A_{n_{2}}$. Then $f|K_{n_{1},m_{1},\beta_{1}}= r_{n_1,\beta_{1}}=r\in C^{1}$ and $k(K_{n_{2},m_{2},\beta_{2}})\subset C_{n_{2},\beta_{2}}$. Now, for $k\in{\cal K}$ observe that $[k=g_{\beta}]\subset\{x_\alpha\colon\alpha<\beta\}$ for each $\beta<\co$, so $\card([k=g_{\beta}])<\co$ and, by Lemma~\ref{lemGDelta}, $k\in SZ$. Therefore, ${\cal K}\subset CIVP\setminus SCIVP$. \Qed \section{An almost continuous S-Z function having the CIVP}\label{sec3} Recall that it is consistent with ZFC that no $SZ$ function $h\colon\real\to\real$ is almost continuous. In fact, this happens in the iterated perfect set model, where there is no $SZ$ function $h\colon\real\to\real$ with the Darboux property~\cite{BCN}. Thus, in this section we shall work with the additional set theoretical assumption. In our construction we will use some ideas from \cite{JC}, \cite{Ke} and \cite{BCN}. We shall need also the following lemma, basic in the theory of almost continuous maps from $\mathR$ into $\mathR$. \begin{Le} \label{blok} {\rm \cite{KG}} If $f\colon\mathR\to\mathR$ intersects every {\em blocking set\/}, i.e., a closed set $K\subset\mathR^2$ with the domain being a non-degenerate interval, then it is almost continuous. \Qed \end{Le} \begin{Th} Assume that the real line is not a union of less than $\co$ many meager sets. Then there exists an almost continuous Sierpi{\'n}ski--Zygmund function which has the CIVP. \end{Th} \pf For $A\subset\mathR$ we denote $L(A)=A\times\mathR$. Let $\{ x_{\alpha}\colon \alpha <\co\}$ be a one-to-one enumeration of $\mathR$, $\{I_n\colon n<\omega\}$ be a sequence of all open intervals with rational end-points, $\{ g_{\alpha}\colon{\alpha <\co}\}$ be an enumeration of all continuous functions defined on $G_{\delta}$ subsets of~$\mathR$ and $\{ P_{\alpha}\colon\alpha<\co\}$ be an enumeration of all Cantor sets. Moreover, let $\{ K_{n,\beta,\gamma}\colon\ n<\omega,\ \beta,\gamma<\co\}$ be a family of pairwise disjoint Cantor sets such that $K_{n,\beta,\gamma}\subset I_n$ for each $n<\omega$ and $\beta,\gamma<\co$. (See, e.g., \cite{KB}.) Let $\varphi\colon\co\to\omega\times\co$ be a bijection and $\varphi=(\varphi_1,\varphi_2)$. Choose, by induction on $\alpha<\co$, a sequence $\la\la C_\alpha,D_\alpha\ra\colon\alpha<\co\ra$ such that for every $\alpha<\co$ \begin{description} \item[(1)] $D_\alpha\subset\dom(g_\alpha)\setminus\bigcup_{\beta<\alpha} (C_\beta\cup D_\beta)$ is at most countable set such that $g_\alpha|D_\alpha$ is a dense subset of $g_{\alpha}\setminus \bigcup_{\beta<\alpha}(g_{\beta}\cup L(C_\beta\cup D_\beta))$; \item[(2)] if $\varphi(\alpha)=(n,\beta)$ then $C_{\alpha}=K_{n,\beta,\gamma}$ for some $\gamma<\co$ with $C_{\alpha}\cap \bigcup_{\delta\leq\alpha}D_{\delta}=\emptyset$. \end{description} The choice as in (2) can be made, since the set $\bigcup_{\delta\leq\alpha}D_{\delta}$ has cardinality less than continuum, and there is continuum many pairwise disjoint sets $K_{n,\beta,\gamma}$, $\gamma<\co$. Now, define the values $f(x_\alpha)$ of function $f$ by induction on $\alpha<\co$ as follows. \begin{description} \item[(a)] $f(x_\alpha)=g_\beta(x_\alpha)$ provided $x_\alpha\in D_\beta$ for some $\beta<\co$. \item[(b)] $f(x_{\alpha})\in P_{\beta}\setminus\{ g_{\delta}(x_{\alpha}) \colon \delta\leq\alpha\}$ provided $x_\alpha\in C_\nu$ and $\varphi_2(\nu)=\beta$. \item[(c)] $f(x_\alpha)\in\mathR \setminus \{g_\gamma(x_\alpha)\colon \gamma\leq\alpha\}$ otherwise. \end{description} We will show that $f$ has the desired properties. To verify that $f$ has the CIVP fix a Cantor set $P\subset\mathR$ and an interval $I\subset\mathR$. There exist $n<\omega$ and $\beta<\co$ such that $I_n\subset I$ and $P=P_\beta$. Let $\alpha =\varphi^{-1}(n,\beta)$. Then $C_\alpha\subset I$ and $f(C_\alpha) \subset P$, so $f\in CIVP$. The proof that $f\in SZ\cap AC$ is the same as in \cite{BCN}. To prove that $f\in SZ$, by Lemma~\ref{lemGDelta} it is enough to show that $\card([f=g_{\beta}])<\co$ for each $\beta <\co$. But $[f=g_{\beta}]\subset \bigcup_{\alpha\leq\beta}D_\alpha\cup\{x_\alpha\colon\alpha<\beta\})$, so $\card([f=g_{\beta}])<\co$. Hence, $f\in SZ$. To verify that $f$ is almost continuous choose a blocking set $F\subset \mathR^2$. By Lemma~\ref{blok}, it is enough to show that $f\cap F\neq\emptyset$. To see this, note that there exist a non-degenerate interval $J\subset\dom(F)$ and an upper semicontinuous function $h\colon J\to\mathR$ such that $h\subset F$. (See \cite[Lemma~1]{Ke}.) Thus there exists an $\alpha_0<\co$ such that $g_{\alpha_0}=h|C(h)$, where $C(h)$ denotes the set of all points at which $h$ is continuous. Then $\dom g_{\alpha_0}$ is residual in $J$ and $g_{\alpha_0}\subset F$. In particular, if $S$ is the set of all $\alpha<\co$ such that $\dom(g_{\alpha}\cap F)$ is residual in some non-degenerate interval $I$ then $S\neq\emptyset$. Let $\alpha=\min S$ and $I$ be a non-degenerate interval such that $\dom(g_{\alpha}\cap F)$ is residual in $I$. But $F$ is closed and $g_\alpha$ is continuous. So, $g_{\alpha}|I\subset F$. Moreover, by the minimality of $\alpha$, for each $\beta<\alpha$ the set $I\cap[g_\beta=g_\alpha]\subset\dom(g_{\beta}\cap F)$ is nowhere dense in $I$. Consequently, \[ I\cap \dom\Bigl [g_\alpha\setminus \bigcup_{\beta<\alpha}(g_{\beta}\cup L(C_\beta\cup D_\beta))\Bigl ]= (I\cap\dom(g_\alpha))\setminus\bigcup_{\beta<\alpha}\Bigl( I\cap([g_\beta=g_\alpha]\cup C_\beta\cup D_\beta)\Bigl)\neq\emptyset, \] since, by our set theoretic assumption, $I$ cannot be cover by less than $\co$ many meager sets . Thus, by (1), $I\cap D_\alpha\neq\emptyset$. Let $x\in I\cap D_\alpha$. Then, by (a), $\la x,f(x)\ra=\la x,g_\alpha(x)\ra\in f\cap F$. \Qed Because no Sierpi{\'n}ski-Zygmund function has the SCIVP, we obtain the following corollary. \begin{Co} Assume that the real line is not a union of less than $\co$ many meager sets. Then there exists an almost continuous function with the CIVP but without the SCIVP. \end{Co} Moreover, because every extendable function has the SCIVP, we have the following result \begin{Co} Assume that the real line is not a union of less than $\co$ many meager sets. Then $$Ext\neq AC+CIVP.$$ \end{Co} Nevertheless, note that the problem whether $Ext=AC+SCIVP$ remains open. (See \cite[Question~4]{RG} or \cite[Question]{RGR}.) We do not know also whether an example of almost continuous function with the CIVP but without the SCIVP can be constructed in ZFC. \begin{thebibliography}{22} \bibitem{BCN} M.~Balcerzak, K.~Ciesielski and T.~Natkaniec, {\it Sierpi{\'n}ski-Zygmund functions that are Darboux, almost continuous or have a perfect road}, submitted. \bibitem{KB} K.~Banaszewski, {\it Algebraic properties of functions with the Cantor Intermediate Value Property}, Math. Slovaca, {\bf 47} (1997), to appear. \bibitem{BCW} A. M. Bruckner, J. G. Ceder, and M. Weiss, {\it Uniform limits of Darboux functions}, Colloq. Math., {\bf 15} (1966), 65--77. \bibitem{JC} J. Ceder, {\it Some examples on continuous restrictions}, Real Anal. Exchange, {\bf 7} (1981--82), 155--162. \bibitem{Da} U. B. Darji, {\it A Sierpi\'{n}ski-Zygmund function which has a perfect road at each point}, Colloq. Math., {\bf 64} (1993), 159--162. \bibitem{RG} R. G. Gibson, {\it Darboux-like functions}, Abstract of the lecture in Banach Center, 1989. \bibitem{CIVP} R. G. Gibson and F. Roush, {\it The Cantor intermediate value property}, Top. Proc., {\bf 7} (1982), 55--62. \bibitem{WCIVP} R. G. Gibson and F. Roush, {\it Concerning the extension of connectivity functions}, Top. Proc., {\bf 10} (1985), 75--82. \bibitem{KK} K. Kuratowski, {\it Topologie I}, Warszawa 1958. \bibitem{KG} K. R. Kellum, {\it Sums and limits of almost continuous functions}, Colloq. Math., {\bf 31} (1974), 125--128. \bibitem{Ke} K. R. Kellum, {\it Almost continuity and connectivity --- sometimes it's as easy to prove a stronger result\/}, Real Anal. Exchange, {\bf 8} (1982--83), 244--252. \bibitem{Max} I. Maximoff, {\it Sur les fonctions ayant la propri\'{e}t\'{e} de Darboux}, Prace Mat.-Fiz., {\bf 43} (1936), 241--265. \bibitem{RGR} H. Rosen, R. G. Gibson and F. Roush, {\it Extendable functions and almost continuous functions with a perfect road}, Real Anal. Exchange, {\bf 17} (1991--92), 248--257. \bibitem{Sh} J. Shoenfield, {\it Martin's axiom\/}, Amer. Math. Monthly, {\bf 82} (1975), 610--617. \bibitem{SZ} W. Sierpi\'{n}ski and A. Zygmund, {\it Sur une fonction qui est discontinue sur tout ensemble de puissance du continu}, Fund. Math., {\bf 4} (1923), 316--318. \bibitem{St} J. Stallings, {\it Fixed point theorems for connectivity maps\/}, Fund. Math., {\bf 47} (1959), 249--263. \end{thebibliography} \end{document}