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%\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),
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