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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
 \thispagestyle{empty}
{
\renewcommand{\thefootnote}{}
 \footnotetext{AMS Subject Classification (1991): Primary: 26A15;
 Secondary: 03E50, 03E65}
}
{
{\renewcommand{\thefootnote}{}
 \footnotetext{Key words:
 Sierpi{\'n}ski-Zygmund functions, almost continuous functions,
 Darboux functions, perfect road, iterated Sacks forcing.}
}



\null\bigskip
\begin{center}
\Large\bf Sierpi\'nski--Zygmund functions that are
 Darboux, almost continuous, or have a perfect road
\end{center}
\bigskip
Marek Balcerzak, Institute of Mathematics,
{\L}\'od\'z Technical University, Al. Politechniki~11,
90--924 {\L}\'od\'z, Poland (mbalce@@krysia.uni.lodz.pl)%
\\[\medskipamount]
Krzysztof Ciesielski, Department of Mathematics, West
 Virginia University, Morgantown, WV 26506--6310, USA
 (kcies@@wvnvms.wvnet.edu)%
 \\[\medskipamount]
 Tomasz Natkaniec, Department of Mathematics, Pedagogical
 University, Chodkiewi\-cza~30, 85-064 Bydgoszcz, Poland
(wspb11@@vm.cc.uni.torun.pl)

\vskip-.5in

%\chKC Nie wyrzucajcie ponizszej komendy \maketitle
% bo sie abstrakt nie drukuje. (To z powodu przestarzalego "style" amsart.)

\maketitle
\begin{abstract}
In this paper we show that if the real line $\mathR$
is not a union of less
than continuum many of its meager subsets
then there exists an almost continuous Sierpi\'nski--Zygmund
function having a perfect road at each point.
We also prove that it is consistent with ZFC that every
Darboux function $f\colon\real\to\real$ is continuous
on some set of cardinality continuum.
In particular, both these results imply that
the existence of a Sierpi\'nski--Zyg\-mund function which is
either Darboux or almost continuous
is independent of ZFC axioms.
This gives a complete solution of a
problem of Darji~\cite{Da}.

The paper contains also a
construction (in ZFC) of an additive
Sierpi\'nski--Zyg\-mund
function with a perfect road at each point.
\end{abstract}


\section{Introduction}
 Our terminology is standard.  In particular, the symbols
 $\mathN$, $\mathZ$, $\mathQ$ and $\mathR$ stand for the sets of
 all: positive integers, integers, rationals and reals,
 respectively.  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 $Y^X$.  The 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 $\{x\in\mathR\colon f(x)=g(x)\}$.

If $\cal J$ is an ideal of subsets of $\mathR$, then
\begin{eqnarray*}
 \cov({\cal J}) &=& \min \{ \card ({\cal F})\colon{\cal
 F}\subset{\cal J}\;\&\;\bigcup{\cal F}=\mathR\}\\
 \unif ({\cal J}) &=& \min\{\card(A)\colon
 A\subset\mathR\;\&\;A\not\in{\cal J}\}.
\end{eqnarray*}
 (See \cite{F}.) The ideal of all meager subsets of $\mathR$ is
 denoted by $\cal K$.  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 bilateral limit point of $C$ and $f|C$ is continuous at $x$.
 We say that $f$ is of {\it perfect road type\/} (shortly, $f\in
 PR$, or $f$ is of PR type) 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) 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 a {\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} if there is a
 connectivity function
 $F\colon \mathR\times [0,1]\to \mathR$ such that
 $F(x,0) = f(x)$ for every $x\in\mathR$.
\end{itemize}

Recall also that if $f\colon\mathR\to\mathR$ intersects every
{\em blocking set\/}, i.e.,
a closed set $K\subset\mathR^2$ whose domain is a
non-degenerate interval, then $f$ is
 almost continuous~\cite{KG}. It is also well-known that each
 almost continuous
 function $f\colon\mathR\to\mathR$ is connected~\cite{St} 
and therefore that it has the Darboux property.

In~\cite{Da}, Darji constructed (in ZFC) an example of an S-Z
function of perfect road type and asked whether
there exists an almost continuous (or just Darboux) S-Z function.
Examples of such functions under additional set theoretical
assumptions are known. For example,
Ceder~\cite{JC} showed that under the assumption of the
Continuum Hypothesis CH there exists a
connectivity (hence Darboux) S-Z function, and
Kellum~\cite{Ke} noticed that Ceder's function is in fact almost
continuous.
In Section~\ref{sec2} we will generalize both constructions
(Ceder's and Darji's) by showing that
under the assumption that $\cov ({\cal K})=\co$
(which is somewhat weaker than CH or
Martin's Axiom~MA~\cite{Sh,F})
there exists an almost continuous S-Z function of PR type.
On the other hand, in Section~\ref{secForc}
we will show that there is a model of ZFC in which there is no
Darboux S-Z function. Thus, some additional set theoretical
assumptions are necessary in all of the examples
mentioned above.

Sections~\ref{sec3} and~\ref{sec4} contain the constructions
related to that from Section~\ref{sec2}.
In particular, Section~\ref{sec3}
deals with the functions $f\colon\mathR\to\mathR$
continuous with respect to the qualitative
topology on the domain and the natural topology on the range.
In Section~\ref{sec4} we
give a ZFC example of an additive S-Z function of PR
type, generalizing the result of Darji from~\cite{Da}.



\section{An almost continuous S-Z function of PR type}\label{sec2}

In our construction we will use the following
easy and well known lemma.

\begin{Le}\label{lemGDelta}
{\rm \cite{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}

The next lemma is a modification of~\cite[Lemma 3]{Da}.

\begin{Le}\label{lemAdd}
 There exists a sequence $\la\la
 H_\alpha,p_\alpha\ra\colon\alpha<\co\ra$
such that
\begin{description}
\item[(1)]
 $H_\alpha\cup\{p_\alpha\}\subset\mathR$ is a compact
 perfect set
 and $p_\alpha$ is a bilaterally limit point
 of~$H_\alpha$;
 \item[(2)]
 $H=\bigcup_{\alpha<\co}H_{\alpha}$ is linearly independent over
 $\mathQ$;
 \item[(3)]
 $H_\alpha\cap H_\beta=\emptyset$ for every $\alpha<\beta<\co$;
\item[(4)]
 for every $x\in\mathR$ there exists continuum many $\gamma<\co$
 such that $x=p_\gamma$.
\end{description}
\end{Le}
 \pf
 Let $K$ be a linearly independent perfect set.
 (See \cite{Jo} or \cite[p.~270]{Ku}.)
Pick a proper perfect subset  $P$ of $K$, and let 
$\{s_{\alpha,n}\colon\alpha<\co\ \&\ n\in\mathZ\setminus\{0\}\}$ 
 be a one-to-one enumeration of $K\setminus P$.
Moreover, let $\{p_\alpha\colon\alpha<\co\}$ be an
 enumeration of $\real$ such that for every $x\in\real$
 there exists continuum many $\gamma<\co$ with
 $p_{\gamma}=x$.
 By induction
on $\alpha<\co$ choose sequences
 $\la q_{\alpha,n}\colon\alpha<\co\ \&\ n\in\mathZ\setminus\{0\}\ra$
 of non-zero rationals and
 $\la C_{\alpha,n}\colon\alpha<\co\ \&\ n\in\mathZ\setminus\{0\}\ra$
 of perfect sets such that for every $\alpha<\co$
and $n\in\mathZ\setminus\{0\}$
 \[
 C_{\alpha,n}\subset\left(p_\alpha,p_\alpha+\frac{1}{n}\right)
 \cap (q_{\alpha,n} s_{\alpha,n} + P),
 \]
 where for $b<a$ we will understand $(a,b)$ as
the interval $(b,a)$.
 Next, for each $\alpha<\co$ define
 $H_{\alpha}=\bigcup\{C_{\alpha,n}\colon\,n\in\mathZ\setminus\{0\}\}$.
 It is easy to see that the family
 $\{H_{\alpha}\subset\real\colon \alpha<\co\}$
 has the desired properties.
 \Qed

\begin{Th}\label{th1}
Assuming $\cov ({\cal K})=\co$, there exists an almost
 continuous S-Z
 function $f\colon\mathR \to\mathR$ which has
 a perfect road at each point.
 \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 $\mathR$ and
 $\{g_{\alpha}\colon{\alpha <\co}\}$ an enumeration of all
 continuous functions defined on $G_{\delta}$ subsets
 of~$\mathR$.

Construct, 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)] $C_\alpha$ is equal to a set $H_\gamma$ from
           Lemma~\ref{lemAdd} such that $x_\alpha=p_\gamma$
           and $C_\alpha$ is disjoint with
$\{x_\beta\colon\beta\leq\alpha\}\cup
\bigcup_{\beta\leq\alpha}D_\beta \cup
\bigcup_{\beta<\alpha}C_\beta$.
\end{description}
 The choice as in (2) can be made, since the set
 $\{x_\beta\colon\beta\leq\alpha\}\cup\bigcup_{\beta\leq\alpha}D_\beta$
 has cardinality less
than continuum, and there are continuum many
 pairwise disjoint sets $H_\gamma$ with $p_\gamma=x_\alpha$.

Now, define the values $f(x_\alpha)$ of the 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)]
 $f(x_\alpha)\in\{y\in\mathR\colon|y-f(x_{\beta})|<|x_{\alpha}-x_{\beta}|\}
\setminus \{g_\gamma(x_\alpha)\colon \gamma\leq\alpha\}$
 provided $x_\alpha\in C_\beta$ for some $\beta<\co$.  (Note that
 $f(x_{\beta})$ is already defined since, by (2),
 $\beta<\alpha$.)
 \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.

First notice that, by (b), $f|(C_{\beta}\cup\{x_\beta\})$ is continuous
 at $x_{\beta}$ for every $\beta <\co$. Therefore, $f\in PR$.

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 $F\subset \mathR^2$. 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 $\cov({\cal K})=\co$. 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

%\vspace{\baselineskip}
\noindent
{\bf Remark.} Note that an S-Z function of perfect road type is
not extendable. (See~\cite{Da}.)
So, Theorem~\ref{th1} gives a new and easy
example of an almost
 continuous function that has a perfect road at each point and is
 not an extendable function. The first example of such a function
 was constructed (in ZFC) in~\cite{GRR}.

 \section{The qualitative case}\label{sec3}
 Now we shall consider $\mathR$ with the fine topology $q$
 generated by the ideal ${\cal K}$. This topology is called the
 {\em qualitative\/} topology. Recall that a set $G$ is open in
 the qualitative topology if it can be written in the form
 $U\setminus P$, where $U$ is open in the Euclidean topology and
 $P$ is of the first category. (Note that it is an example of a
 $^\ast$-topology in the sense of Hashimoto \cite{H} or $\cal
 J$-topology in the sense of Vaidyanathaswamy \cite{V} with
 respect to the ideal $\cal K$ of meager sets.)

For a set $A\subset \mathR$ and a function $f\colon A\to \mathR$
 we say that $f$ is {\em $q$-continuous} at a point $x_0\in A$ if
 $f$ is continuous at $x_0$ as a real function defined on the
 subspace $A$ of the space $\la\mathR,q\ra$.

\begin{Le}\label{LemQeal}
 For every set $A\subset\mathR$ and a function $f\colon A\to\mathR$,
\begin{enumerate}
 \item
 if $A\in {\cal K}$, then $f$ is $q$-continuous;
\item
 if $f$ is continuous, then it is $q$-continuous;
 \item
 if $A$ is $q$-dense in itself and $f$ is $q$-continuous, then
 $f$ is continuous.
 \end{enumerate}
 \end{Le}
\pf
Statements (1) and (2) are evident. For (3), see %e.g.
\cite[Th.~4]{M} or~\cite[Cor.~1.1.8]{CLO}.
\Qed

\begin{Pn}
If $\cov ({\cal K})=\unif({\cal K})=\co$ then there
exists an almost continuous function
$f\colon\mathR\to\mathR$
of perfect road type such that
 $f|M$ is not $q$-continuous for every $M\not\in{\cal K}$.
\end{Pn}
\pf
 Assume $M\not\in{\cal K}$. Then $M$ is not $q$-nowhere dense. So, 
 there exists an
 interval $I$ such that $M\cap I$ is $q$-dense in itself.
 In particular, $M\cap I\not\in{\cal K}$, so
 $\card (M\cap I)=\co$.
 Let $f$ be the function constructed in
 Theorem~\ref{th1}. Then $f|(M\cap I)$ is discontinuous
 so, by Lemma~\ref{LemQeal}, $f|(M\cap I)$
 is not $q$-continuous. Therefore, $f|M$ also
 is not $q$-continuous.
 \Qed

\section{An additive S-Z function of PR type}\label{sec4}

\begin{Th}
 There exists an additive Sierpi\'nski--Zygmund
 function $f\colon \mathR\to\mathR$ of perfect road type.
\end{Th}
 \pf 
 Let $\widehat{H}=\{h_{\alpha}\colon \alpha <\co\}$ be a Hamel
 basis which
 contains the set $H$ constructed in
 Lemma~\ref{lemAdd} and let
 $\{g_{\alpha}\colon \alpha <\co\}$ be a
 well-ordering of all continuous functions defined on
 $G_{\delta}$ subsets of $\mathR$. For each $\alpha<\co$ choose a
 set $\widehat{H}_{\alpha}$ and an
$\hat f( h_{\alpha})$ such that
\begin{description}
\item[(a)] $\widehat{H}_{\alpha}$ is equal to a set $H_{\gamma}$ from
    Lemma~\ref{lemAdd} such that $h_{\alpha}=p_{\gamma}$ and
    $\widehat{H}_{\alpha}$ is disjoint from
    $\{h_{\beta}\colon\beta\leq\alpha\}$;
\item[(b)]
    $\hat f(h_{\alpha})\neq q g_{\beta}(x)-f_\alpha(t)$ for all
    $\beta \leq\alpha$, $q\in\mathQ$,
    $x\in \lin (\{ h_{\beta}\colon \beta \leq \alpha\})$
    and $t\in \lin (\{ h_{\beta}\colon \beta < \alpha\})$,
    where $\lin (A)$ denotes the linear subspace of
    $\mathR$ over $\mathQ$ generated by $A$, and
    $f_\alpha$ is the additive extension of
    $\hat f|\{ h_{\beta}\colon \beta <\alpha\}$.
\end{description}
 Moreover, if $h_{\alpha}\in \widehat{H}_{\beta}$ for some
  $\beta<\co$ then, by (a),
  $\beta<\alpha$ and we will additionally require that
 \begin{description}
 \item[(c)]
 $|\hat f(h_{\alpha})-\hat f(h_{\beta})|\leq |h_{\alpha}-h_{\beta}|$.
 \end{description}

 Let $f\colon \mathR\to\mathR$ be the additive extension of
 $\hat f\colon \widehat{H}\to\real$.

To prove that $f$ is a function of S-Z type it is enough to
verify that
$\card (f\cap g_{\alpha})<\co$ for every $\alpha<\co$. So, fix
$\alpha<\co$ and
 assume that $f(x)=g_{\alpha}(x)$.  Let $\gamma$ be the first
 ordinal such that
 $x\in \lin (\{ h_{\beta}\colon\beta\leq\gamma\})$.
 Then $x=ph_{\gamma}+t_0$, where $p\in\mathQ\setminus\{ 0\}$ and
 $t_0\in \lin (\{ h_{\beta}\colon \beta<\gamma\})$. So
 $h_{\gamma}=qx - t$, where $q=p^{-1}\in\mathQ$ and
 $t=qt_0\in\lin (\{ h_{\beta}\colon\beta <\gamma\})$.
 Moreover, $\hat f(h_{\gamma})=f(h_{\gamma})=
 qf(x)-f(t)=qg_{\alpha}(x)-f(t)$, so $\gamma<\alpha$.
Thus, by (b),
 $[f=g_{\alpha}]\subset \lin (\{ h_{\beta}\colon \beta<\alpha\})$
 and $\card (f\cap g_{\alpha})<\co$.

Now we shall verify that $f$ has a perfect road at each $x\in\mathR$.
For $x=h_{\alpha}\in\widehat{H}$ it is obvious  by (c),
since $f|(\widehat{H}_{\alpha}\cup\{h_{\alpha}\})$ is continuous at
 $h_{\alpha}$. So, assume
 that $x=\sum_{i=1}^nq_ih_{\alpha_i}$, where all $q_i$ are
 rationals. Then $x$ is a bilaterally limit point of a perfect
 set $\widehat{H}_x=\sum_{i=1}^n q_i\widehat{H}_{\alpha_i}\cup\{
 x\}$ and $f|\widehat{H}_x$ is continuous at $x$.
\Qed

\section{A model with no Darboux S-Z function}\label{secForc}

In this section we will show that in the iterated perfect set
 (Sacks) model there is no Darboux Sierpi\'{n}ski-Zygmund
 function.  We will describe here only those properties of this
 model that are necessary to follow the argument.  More details
 can be found in~\cite{Mi} and~\cite{BL}.

Let $V$ be a model of ZFC+CH and let $V[G_{\omega_2}]$ be a model
 of ZFC+$\co=\omega_2$ obtained as a generic extension of $V$
 over the forcing $\poset$, which is a countable support
 iteration of the perfect set (Sacks) forcing.  Then $V$ and
 $V[G_{\omega_2}]$ have the same cardinals. Moreover, in
 $V[G_{\omega_2}]$ there exists an increasing sequence $\la
 V[G_\alpha]\colon \alpha\leq\omega_2\ra$ (of proper classes in
 $V[G_{\omega_2}]$, given by a formula) with the following
 properties. ($V[G_\alpha]$ is a generic extension of $V$
 obtained by extending $V$ with the part $G_\alpha$ of
 $G_{\omega_2}$ which belongs to the $\alpha$-iteration of Sacks
 forcing.)

\begin{description}
\item[(A)] CH holds in $V[G_\alpha]$ for every $\alpha<\omega_2$.
\item[(B)] For every $\alpha<\omega_2$ of uncountable cofinality
and every $s\in 2^\omega\cap V[G_\alpha]$ there exists $\beta<\alpha$
such that $s\in V[G_\beta]$.
\item[(C)] For every $\alpha<\omega_2$ and $a,b\in\real$, $a<b$ ,
there exists
$s\in(a,b)\cap(V[G_{\omega_2}]\setminus V[G_\alpha])$
(a Sacks number over $V[G_\alpha]$) such that for every
$x\in\real\cap(V[G_{\omega_2}]\setminus V[G_\alpha])$
there exists
 a continuous function
$g\in\real^\real\cap V[G_{\omega_2}]$ coded in
$V[G_\alpha]$ (i.e., such that $g|\mathQ\in V[G_\alpha]$)
with the property that $g(x)=s$.
\end{description}
Property (A) follows immediately from the fact that CH holds in $V$
and we iterate forcings of cardinality $\co$.
Properties (B) and (C) can be found in~\cite[Thm. 3.3(a)]{BL}
and in~\cite[Sec.~4, p.~581]{Mi}, respectively.

Note also, that property (B) can be modified as follows.
\begin{description}
\item[(B$^\prime$)] For every $\alpha<\omega_2$ of uncountable cofinality
and every $p\in(\real^\mathQ\cup\real)\cap V[G_\alpha]$ there exists
$\beta<\alpha$ such that $p\in V[G_\beta]$.
\end{description}
The part concerning $p\in\real$ follows from the fact that
a real number can be identified with
its binary representation, i.e., a function $s\colon\omega\to 2$.
This also implies the part for $p\in\real^\mathQ$, since
any such $p$ can be identified with
$\hat p\colon\mathQ\times\omega\to 2$, $\hat p(q,n)=p(q)(n)$,
and further, with a function from $2^\omega$
by identifying $\mathQ\times\omega$ with $\omega$ via bijection
from $V$.

Now, let $h\in \real^\real\cap V[G_{\omega_2}]$ be an
SZ function and let $a=\inf h[\real]$, $b=\sup h[\real]$.
Then $-\infty\leq a<b\leq\infty$.
We will show that $(a,b)\not\subset h[\real]$.

To prove this
let $C(\real)$ stand for the set of all continuous functions
from $\real$ to $\real$ and define,
for $\beta<\omega_2$,
\[
S_\beta=
%\Bigl((\real\cup\real^\mathQ)\cap V[G_\beta]\Bigl)\cup
%\TN1
h\Bigl [\real\cap V[G_\beta]\Bigl ]\cup
\bigcup \Bigl \{\{x,y\}\colon
(\exists g\in C(\real)\cap V[G_{\omega_2}])
(g|\mathQ\in V[G_\beta]\; \&\; \la x,y\ra\in g\cap h) \Bigl \}.
\]
 Note that, by (A), the set $(\real\cup\real^\mathQ)\cap
 V[G_\beta]$ has cardinality $\leq\omega_1$ and that $\card(h\cap
 g)\leq\omega_1$ for every $g\in C(\real)\cap V[G_{\omega_2}]$.
 So, $\card(S_\beta)\leq\omega_1$ for every $\beta<\omega_2$.
 Define $\Gamma\colon\omega_2\to\omega_2$ by putting
 $\Gamma(\beta)=\sup\{\gamma(x)\colon x\in S_\beta\}$,
 where $\gamma(x)=\min\{\beta\colon\, x\in V[G_{\beta}]\}$,
and let $\alpha<\omega_2$ be of uncountable
cofinality such that $\Gamma(\beta)<\alpha$
for every $\beta<\alpha$.
Then, by (B$^\prime$),
\begin{description}
 \item[(i)] $h(x)\in V[G_\alpha]$ for every $x\in\real\cap
            V[G_\alpha]$;
\item[(ii)] $h\cap g\subset V[G_\alpha]$ for every
            $g\in C(\real)$ coded in $V[G_\alpha]$.
\end{description}

Now, let $s\in(a,b)\cap(V[G_{\omega_2}]\setminus V[G_\alpha])$
be a number from (C).
It is enough to prove that $s\not\in h[\real]$.

But $s\not\in h\Bigl [\real\cap V[G_\alpha]\Bigl ]$ by (i).
So, let $x\in\real\cap (V[G_{\omega_2}]\setminus V[G_\alpha])$.
It is enough to show that $h(x)\neq s$.
But, by (C), there exists a continuous function
$g\colon\real\to\real$ coded in $V[G_\alpha]$
such that $g(x)=s$. So, $h(x)\neq s$, since otherwise
$\la x,s\ra\in h\cap g$ and, by (ii), $s\in V[G_\alpha]$.
This contradiction finishes the proof. \Qed


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