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\documentclass{rae}
\usepackage{amsmath,amsthm,amssymb}
%\coverauthor{Krzysztof Banaszewski and Tomasz Natkaniec}
%\covertitle{Sierpi\'nski--Zygmund Functions that have the Cantor Intermediate Value Property}
\received{May 5, 1998}
\MathReviews{Primary 26A15.}
\keywords{ Sierpi{\'n}ski-Zygmund functions, almost continuous
functions, extendable functions, the Cantor intermediate value
property CIVP, the strong Cantor intermediate value property
SCIVP.}
\firstpagenumber{1}
\markboth{K. Banaszewski and T. Natkaniec}{Sierpi\'nski--Zygmund functions with CIVP}
\author{
Krzysztof Banaszewski, Department of Mathematics, Pedagogical
University, Chodkiewi\-cza~30, 85-064 Bydgoszcz, Poland,
e-mail: {\tt krzysb@zachem.com.pl}\\
Tomasz Natkaniec\thanks{This author was partially supported 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
e-mail: {\tt mattn@ksinet.univ.gda.pl}}
\title{SIERPI\'NSKI--ZYGMUND FUNCTIONS THAT HAVE
THE CANTOR INTERMEDIATE VALUE PROPERTY}
%%%%%%Put Author's Definitions Below Here %%%%%%%%%%%
\newcommand{\bd}{\rm bd}
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\begin{document}
\maketitle
\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}
\bigskip
\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 $f,g\in\mathR^{\mathR}$ 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 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 \cite{RG} or the survey \cite{GN}.) 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 generalizes 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}
Let ${\cal U}_0$ be the class of all functions $f$ such that for every
interval $J$ the set $f(J)$ is dense in the interval
$[\inf_{J}f,\sup_{J}f]$ \cite{BCW}.
\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 $m\in \mathbb{N}$.
\end{Le}
\begin{Th}
There exists a Sierpi{\'n}ski--Zygmund function having the CIVP.
\end{Th}
\begin{proof}[{\sc Proof.}]
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, $f\in SZ$.
\end{proof}
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
${\cal U}\subset {\cal U}_0$~\cite{BCW} and ${\cal U}\cap WCIVP={\cal
U}\cap PR$.)
\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}
\begin{proof}[{\sc Proof.}]
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 consequently, $f\in SCIVP$.
Now, for $k\in{\cal K}$ observe that
$k(K_{n_{2},m_{2},\beta_{2}})\subset C_{n_{2},\beta_{2}}$ and
$[k=g_{\beta}]\subset\{x_\alpha\colon\alpha<\beta\}$ for each
$\beta<\co$, so $\card([k=g_{\beta}])<\co$ and, by
Lemma~\ref{lemGDelta}, $k\in SZ$. Therefore, $k\in CIVP\setminus
SCIVP$.
\end{proof}
\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
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}
\begin{proof}[{\sc Proof.}]
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 an 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 closed set $F\subset \mathR^2$ with the domain being a
non-degenerate interval. 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,
\begin{eqnarray*}
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,
\end{eqnarray*}
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), $\la x,f(x)\ra=\la x,g_\alpha(x)\ra\in f\cap F$.
\end{proof}
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.
\section{Final remarks}
The following new results concerning the problems above were
obtained after this paper was written.
\begin{itemize}
\item
Krzysztof Ciesielski constructed in ZFC an example of an
additive function in the class $AC+CIVP\setminus
SCIVP$~\cite{Add}.
\item
Harvey Rosen proved that CH implies the inequality
$AC+SCIVP\setminus Ext\neq\emptyset$~\cite{HR}.
The same result was recently obtained in ZFC by Krzysztof
Ciesielski and Andrzej Ros{\l}anowski~\cite{CRos}.
\end{itemize}
\begin{thebibliography}{22}
\bibitem{BCN}
M.~Balcerzak, K.~Ciesielski and T.~Natkaniec, {\it
Sierpi{\'n}ski-Zygmund functions that are Darboux, almost
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\end{document}
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{Sierpi\'nski--Zygmund Functions that have the Cantor Intermediate Value Property}
{Krzysztof Banaszewski and Tomasz Natkaniec}
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