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\documentclass{proc-l} \issueinfo{127}{12}{December}{1999}
\pagespan{3615}{3622}
\dateposted{May 13, 1999}
\PII{S 0002-9939(99)04955-2}
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\begin{document}
\title[Decomposing symmetrically continuous functions]
{Decomposing symmetrically continuous\\
and
Sierpi\'nski-Zygmund functions\\
into continuous functions}
\author{Krzysztof Ciesielski}
\commby{Alan Dow}
\date{November 23, 1997 and, in revised form, February 18, 1998}
\thanks{The author was partially supported by
NATO Collaborative Research Grant CRG~950347 and 1996/97 West Virginia
University Senate Research Grant.}
\subjclass{Primary 26A15;
Secondary  03E35}
\keywords{Decomposition number,
symmetrically continuous functions, Sierpi\'nski-Zygmund functions} \address{Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506-6310} \email{KCies@wvnvms.wvnet.edu} 

\begin{abstract}
In this paper we will investigate the smallest cardinal number
$\kappa$ such that for any symmetrically continuous function
$f\colon\mathbb{R}\to\mathbb{R}$ there is a partition
$\{X_\xi\colon\xi<\kappa\}$
of $\mathbb{R}$ such that every restriction
$f\restriction X_\xi\colon X_\xi\to\mathbb{R}$
is continuous. The similar numbers for the classes
of Sierpi\'nski-Zygmund functions and all functions from $\mathbb{R}$ to
$\mathbb{R}$
are also investigated and it is proved that all these numbers are equal.
We also show that $\mathrm{cf}(\mathfrak{c})\leq\kappa\leq\mathfrak{c}$
and that
it is consistent with ZFC that each of these inequalities is strict.
\end{abstract}

\maketitle

\section{Preliminaries}
Our notation and terminology is standard and follows \cite{Ci}.
In particular, $|X|$ will stand for the cardinality of $X$.
For a cardinal number $\kappa$ we will write $\cf(\kappa)$
for its cofinality. We also define
$[X]^{\kappa}=\{Y\subseteq X\colon |Y|=\kappa\}$.
The definition of $[X]^{<\kappa}$ is similar.
The cardinality of the set $\real$ of real numbers is denoted by~$\co$.
The functions are identified with their graphs.
The class of all function from a set $X$ into a set $Y$ is denoted by $Y^X$.

For $Z\subset\real$ and
a cardinal number $\kappa\leq\co$ let $\Pi_\kappa(Z)$
denote the family of all coverings of $Z$ with at most $\kappa$
many sets. We will write $\Pi_\kappa$ for $\Pi_\kappa(\real)$.
In~\cite{cimopaso} the authors considered the following
cardinal {\em decomposition} function for arbitrary
families $\sF\subset\real^Z$, with $Z\subset\real$,
and $\sG\subset\bigcup\{\real^X\colon X\subset Z\}$:
\[
\dec(\sF,\sG)\!=\min(\{\kappa\leq\co\colon(\forall f\in\sF)
(\exists \mathcal{X}\!\in\Pi_\kappa(Z))
(\forall X\in \mathcal{ X})(f\!\restriction\! X\in\sG)\}\cup\{\co^+\}).
\]
In particular, if $\sC$ stands for the family of all
continuous functions from a subset of $\real$ into $\real$, then
\[
\text{$f\colon\real\to\real$ is countable continuous if and only if
$\dec(\{f\},\sC)\leq\omega$.}
\]
In ~\cite{cimopaso} the authors considered the values
of $\dec(\sB_\beta,\sB_\alpha)$ for $\alpha<\beta<\omega_1$, where
$\sB_\alpha$ stands for the functions of $\alpha$-th Baire class.
In particular, they proved that
\[
\mathrm{cov}(\sM)\leq \dec(\sB_1,\sC)\leq d,
\]
where $\mathrm{cov}(\sM)$ is the smallest cardinality of
a covering of $\real$ by
meager sets, and $d$, the dominating number, is the smallest cardinality
of a dominating family $D\subset\omega^\omega$, that is, such that
for every $f\in\omega^\omega$ there exists $g\in D$ such that
$f\leq^* g$.
Moreover, in papers
\cite{step.30} and \cite{step.34} it has been proved that
each of these inequalities can be strict.

There are also some interesting results concerning
the value of $\dec(\sC,\sD)$, where $\sD$ is the class of all (partial)
differentiable functions. It has been proved by Morayne
\cite[Theorem~6.1]{step.38} that
$\dec(\sC,\sD)\geq \mathrm{ cov}(\sM)$, while
Stepr\={a}ns \cite{step.38} proved
that it is consistent with ZFC that
$\dec(\sC,\sD)<\co$.

For more information on the subject see
also a survey paper~\cite{KC:STAsurvey}.



In this paper we will examine the numbers $\dec(\sF,\sC)$,
where $\sF$ is one of the following three classes:
\begin{enumerate}
\item[]{$\real^\real$} of all functions from $\real$ to $\real$;
\item[]{$\sz(X)$} of all {\em Sierpi\'nski-Zygmund}
functions from $X\subseteq\real$ into $\real$, that is, all
$\quad f\colon X\to\real$ whose restrictions $f\restriction Y$
are discontinuous for all subsets $Y$ of $X$\quad
of cardinality continuum $\co$
(we will write $\sz$ for $\sz(\real)$);
\item[]{$\Sc$} of all {\em symmetrically continuous}
functions $f\colon\real\to\real$, that is, such that for 
\quad every
$x\in\real$ we have
\[
\lim_{h\to 0^+}f(x-h)-f(x+h)=0.
\]
\end{enumerate}
Now, since $\sF\subset\sF'$ implies $\dec(\sF,\sG)\leq\dec(\sF',\sG)$
and a Sierpi\'nski-Zygmund
function $f\colon X\to\real$, with  $X\in[\real]^\co$,
cannot be covered by less than
$\cf(\co)$-many continuous functions, we obtain immediately
the following inequalities:
\begin{equation}\label{eq1}
\cf(\co)\leq \dec(\{f\},\sC)\leq \dec(\sz,\sC)\leq \dec(\real^\real,\sC)\leq \co
\end{equation}
for any $f\in\sz(X)$ with $X\in[\real]^\co$. Also, in~\cite{CSz}
the authors proved $\dec(\Sc,\sC)>\omega$ by showing that
there exists an $f\in\Sc$ such that $f\restriction X\in\sz(X)$ for some
$X\in[\real]^\co$. This clearly implies that
\begin{equation}\label{eq2}
\cf(\co)\leq \dec(\{f\restriction X\},\sC)\leq \dec(\Sc,\sC)\leq
\dec(\real^\real,\sC)\leq \co.
\end{equation}
The main results of this paper are the following three theorems
refining the results of (\ref{eq1}) and (\ref{eq2}).

\begin{theorem}[Ciesielski, Szyszkowski]\label{th1}
$\dec(\Sc,\sC)=\dec(\real^\real,\sC)$.
\footnote{This 
result was obtained during Mr. Szyszkowski's
work on his
doctoral degree under the author's supervision.
It will most likely become a part of Mr. Szyszkowski's dissertation.}
\end{theorem}
\begin{theorem}\label{th1a} {$\dec(\sz,\sC)=\dec(\real^\real,\sC)$.}
\end{theorem}

Theorems~\ref{th1} and~\ref{th1a} immediately imply the following corollary.

\cor{cor1}{
$\cf(\co)\leq\dec(\sz,\sC)=\dec(\Sc,\sC)=\dec(\real^\real,\sC)\leq\co$.}

The next theorem tells us that
neither of the inequalities in Corollary~\ref{cor1}
can be replaced by the equation.

\begin{theorem}\label{th2} {Let $\lambda$ be a cardinal number with uncountable cofinality.
Then
\begin{enumerate}
\item[(a)] it is relatively consistent with ZFC that
$\co=\lambda=\dec(\real^\real,\sC)$; and
\item[(b)] it is relatively consistent with ZFC that
$\co=\lambda$ and $\dec(\real^\real,\sC)=\cf(\lambda)$.
\end{enumerate}
}\end{theorem}

The statement of the theorem is not very precise.
(For example, we can't have $\kappa=\co^+$
in the conclusion of the theorem.)
Our interpretation of it is that if $\lambda$
satisfies the assumption of the theorem in a model $M$ of ZFC
satisfying the Generalized Continuum Hypothesis GCH,
then we can find an extension of $M$ to another ZFC model
in which all the cardinals are the same, have the same cofinalities, and
either (a) or (b) holds.
In particular Theorem~\ref{th2} implies the following corollary.

\cor{cor2}{
\textup{(a)} It is consistent with ZFC that $\cf(\co)<\co=\dec(\real^\real,\sC)$.
\par
\textup{(b)} It is consistent with ZFC that $\dec(\real^\real,\sC)=\cf(\co)<\co$.
}

Theorems~\ref{th1}, \ref{th1a}, and~\ref{th2}
will be proved in the next sections.
We will finish this section with the following open problem.

\pr{prob1}{Is it consistent with ZFC that $\cf(\co)<\dec(\real^\real,\sC)<\co$?}



\section{Proof of Theorem~\ref{th1}} 

In order to prove the theorem we will show the following fact that is
of independent interest.

\prop{prop1}{There exists a perfect set $P\subset\real$ with the property that
for every $f_0\colon P\to[0,1]$ there is a symmetrically
continuous function $f\colon\real\to\real$ extending $f_0$.}

The proof is a compilation of the results contained
in~\cite{CSz} and~\cite{Ch}. In particular, if
$C(f)$ stands for the set of points of continuity of
a function $f\colon\real\to\real$ and $D(f)=\real\setminus C(f)$,
then the following lemma is a rephrasing of \cite[Lemma~2.1]{CSz}
(used with $\{A_\alpha\}_{\alpha\in \mathcal{ A}}=\{2D(h)+y\colon y\in X\}$).

\lem{lemA}{Let $h\colon\real\to\real$ and $X\subset D(h)$
be such that $h$ is symmetrically continuous and
\begin{enumerate}
\item[(i)]   $C(h)=h^{-1}(0)$,
\item[(ii)] $D(h)$ is an additive subgroup of $\real$,
\item[(iii)]
 $(2D(h)+x)\cap(2D(h)+y)=\emptyset$ for every distinct $x,y\in X$.
\end{enumerate}
Then for every map $r\colon X\to[0,1]$ the function
\[
f(x)=h(x)\cdot\sum_{y\in X} r(y) \charf{2D(h)+y}(x)
\]
is symmetrically continuous.}

Now, in~\cite{Ch} (in the proof of Theorem~1)
Chleb\'{\i}k shows that the function
$h\colon\real\to[0,1]$ defined by formula
\[
h(x)=\lim_{m\to\infty}\left(
1+\sum_{n=1}^m \left|\frac{1}{n}\sin 3^n x\right|
\right)^{-1}
\]
is upper semicontinuous,
symmetrically continuous, and satisfies
(i) and (ii) from Lemma~\ref{lemA}.
Towards the construction of a set $X$ he defines
the following.

He takes an arbitrary
linear basis $\sH\subset(0,1]$ of $\real$ over $\rational$
with $1\in\sH$, puts $\Lambda=\sH\setminus\{1\}$,
and defines $X=\pi\cdot\psi[\Lambda]$
for a continuous injection
$\psi\colon(0,1]\setminus\rational\to(0,1)$
given by the formula
\[
\psi(x)=\sum_{k=1}^\infty \mu_k\cdot 3^{-(2^k+1)},
\]
where $x=\sum_{k=1}^\infty \mu_k\cdot 2^{-k}$,
with $\mu_k\in\{0,1\}$, is the unique
binary representation of $x$.
Chleb\'{\i}k proves also that $X\subset D(h)$ and that
\[
2D(h)+H_1\neq 2D(h)+H_2\ \ \ \text{ for every distinct
$H_1,H_2\subset X$},
\]
while this last property is used in~\cite[Lemma~2.4]{CSz}
to prove that $X$ satisfies (iii) of Lemma~\ref{lemA}.

Thus, $h$ and $X$ satisfy the assumptions of Lemma~\ref{lemA}.
However, to prove Proposition~\ref{prop1} we need also
two additional facts that
\begin{enumerate}
\item[(iv)] $h$ is of Baire class one (as upper semicontinuous),  and that
\item[(v)] $X$ is a continuous image of $\sH\setminus\{1\}$,
where $\sH$ is an arbitrary linear basis of $\real$ over $\rational$
with $1\in\sH\subset(0,1]$.
\end{enumerate}

Now, take a perfect set
$K\subset[0,1]$ which is linearly independent over $\rational$.
(See e.g. \cite[Theorem~2, Ch.~XI sec.~7]{Kucz}.)
Decreasing it, if necessary, we can assume that $1$ is linearly independent of
$K$. (If it is not, take a finite subset $A$ of $K$ which spans $1$ and
replace $K$ by its perfect subset disjoint with $A$.)
Thus, there exists
a linear basis $\sH\subset(0,1]$ of $\real$ over $\rational$
such that $\{1\}\cup K\subset \sH$.
In particular, $X_0=\psi[K]$ is a perfect subset of
$X=\psi[\sH\setminus\{1\}]$, so it satisfies (iii).

Now, $h\restriction X_0$ has a point of continuity, say $x_0\in X_0$,
since $h$ is of Baire class one. (See e.g. \cite{Br}.)
Since $x_0\in X\subset D(h)=\real\setminus h^{-1}(0)$,
we have $h(x_0)\neq 0$. Thus, we can take a perfect subset $P$ of $X_0$ for
which
$h[P]\subset [b,1]$ for some $b>0$.
We will show that $P$ satisfies Proposition~\ref{prop1}.

To see it note that by Lemma~\ref{lemA} used with $X=P$ the function
\[
f(x)=\frac{1}{b}h(x)\cdot\sum_{y\in P} r(y) \charf{2D(h)+y}(x)
\]
is symmetrically continuous
for any function $r\colon P\to[0,1]$. Moreover,
$f(x)=\frac{1}{b}h(x)\cdot r(x)$ for every $x\in P$
since $x\in 2D(h)+x$. ($D(h)$ is a group.)
Thus defining $r$ by
\[
r(x)=\frac{b}{h(x)}\cdot f_0(x)\leq f_0(x)
\]
we obtain that $r\colon P\to[0,1]$ and
\[
f(x)=\frac{1}{b}h(x)\cdot r(x)=f_0(x)
\]
for every $x\in P$. This finishes the proof of Proposition~\ref{prop1}.



\begin{proof}[Proof of Theorem~\ref{th1}]
Clearly $\dec(\Sc,\sC)\leq\dec(\real^\real,\sC)$.
To prove the other inequality take an
arbitrary $g\in\real^\real$ and let $\kappa=\dec(\{g\},\sC)$.
It is enough to prove that
\begin{equation}\label{eqAA}
\kappa\leq\dec(\Sc,\sC).
\end{equation}

If $\kappa<\omega_1$, then (\ref{eqAA}) follows from
$\cf(\co)\leq\dec(\Sc,\sC)$. So, we can assume that
$\kappa$ is uncountable.
Also, if $h$ is a homeomorphism between $\real$ and $(0,1)$,
then it is easy to see that
$\dec(\{g\},\sC)=\dec(\{h\circ g\},\sC)$. So, we can assume that
$g\colon\real\to(0,1)$.
Moreover, if $\sN=\real\setminus\rational$, then
$\kappa=\dec(\{g\},\sC)=\dec(\{g\restriction \sN\},\sC)$,
since $\kappa$ is uncountable.

Let $h$ be a homeomorphism between $\sN$ and a subset $M$ of $P$,
where $P$ is from Proposition~\ref{prop1},
and define $f_0\colon M\to[0,1]$ by $f_0=g\circ h^{-1}$.
Once again it is easy to see that
$\kappa=\dec(\{g\},\sC)=\dec(\{f_0\},\sC)$.
Now, if $f$ is a symmetrically continuous function
extending $f_0$, which exists by Proposition~\ref{prop1},
then
\[
\kappa=\dec(\{f_0\},\sC)\leq\dec(\{f\},\sC)\leq\dec(\Sc,\sC)
\]
proving (\ref{eqAA}). The proof of Theorem~\ref{th1} is complete.
\renewcommand{\qed}{}\end{proof}


\section{Proof of Theorem~\ref{th1a}} 

The inequality
$\dec(\sz,\sC)\leq\dec(\real^\real,\sC)$ follows from (\ref{eq1}).
To prove the other inequality let $\kappa=\dec(\real^\real,\sC)$.
We will prove that
\begin{equation}\label{eqFcc}
\kappa\leq\dec(\sz,\sC).
\end{equation}

First note that there exists an $f\in\real^\real$ such that
\begin{equation}\label{eqFdd}
\dec(\{f\},\sC)=\dec(\real^\real,\sC)=\kappa.
\end{equation}
Indeed, if $\kappa$ is a successor cardinal, then (\ref{eqFdd}) is obvious.
So assume that $\kappa$ is a limit cardinal.
Clearly
for every $\xi<\kappa$ there exists an $f_\xi\colon\real\to\real$ such that
$\dec(\{f_\xi\},\sC)\geq|\xi|$.
Then $\dec(\{f_\xi\},\sC)=\dec(\{f_\xi\restriction\sN\},\sC)$
for every $\omega_1\leq\xi<\lambda$, where
$\sN=\real\setminus\rational$.
Take a family $\{\sN_\xi\colon\xi<\kappa\}$ of
pairwise disjoint subsets of $\real$ homeomorphic to $\sN$
and let $h_\xi\colon\sN_\xi\to\sN$ be the homeomorphisms.
It is easy to see that
$\dec(\{f_\xi\restriction\sN\},\sC)=\dec(f_\xi\circ h_\xi,\sC)$.
Thus, if $f\in\real^\real$ is any extension of
$\bigcup_{\xi<\kappa}f_\xi\circ h_\xi$, then
\[
\kappa=\sup_{\omega_1\leq\xi<\kappa}|\xi|\leq
\sup_{\omega_1\leq\xi<\kappa}\dec(\{f_\xi\},\sC)\leq\dec(\{f\},\sC)
\leq\dec(\real^\real,\sC)=\kappa
\]
proving (\ref{eqFdd}).

Now, if $\kappa\leq\cf(\co)$, then (\ref{eqFcc})
follows from (\ref{eq1}). So, we will be assuming that
\begin{equation}\label{eqFccc}
\kappa>\cf(\co).
\end{equation}
Then, by (\ref{eq1}), $\cf(\co)<\co$.

Let $f\in\real^\real$ be such that
$\dec(\{f\},\sC)=\kappa$ and let
$\{X_\xi\colon \xi<\cf(\co)\}$ be a partition of
$\real$ such that $|X_\xi|<\co$ for every $\xi<\cf(\co)$.
Notice that
\begin{equation}\label{eqFee}
\dec(\{f\},\sC)\leq\sup_{\xi<\cf(\co)}\dec(\{f\restriction X_\xi\},\sC).
\end{equation}
To see it let $\lambda=\sup_{\xi<\cf(\co)}\dec(\{f\restriction X_\xi\},\sC)$
and for every $\xi<\cf(\co)$
choose $\sX_\xi\in\Pi_{\lambda}(X_\xi)$
such that $f\restriction X\in\sC$ for every $X\in \sX_\xi$.
Then the family $\sX=\bigcup_{\xi<\cf(\co)}\sX_\xi$ has cardinality
at most $\cf(\co)\otimes\lambda$. Therefore,
$\kappa=\dec(\{f\},\sC)\leq\cf(\co)\otimes\lambda$.
Hence, by (\ref{eqFccc}), $\lambda\geq\kappa=\dec(\{f\},\sC)$
proving (\ref{eqFee}).

To finish the proof, let $\{g_\xi\colon\xi<\co\}$ be an enumeration
of all continuous functions from a $G_\delta$ subset of $\real$
into $\real$ and let $\la\lambda_\xi\colon\xi<\cf(\co)\ran$
be an increasing sequence cofinal with $\co$.
For every $\xi<\cf(\co)$ choose a number
$b_\xi\in\real$ such that
\begin{equation}\label{eqFff}
(b_\xi + f[X_\xi])\cap\bigcup_{\zeta<\lambda_\xi}g_\zeta[X_\xi]=\emptyset.
\end{equation}
Such a number can be found since the sets
$f[X_\xi]$ and $\bigcup_{\zeta<\lambda_\xi}g_\zeta[X_\xi]$ have cardinality
less than $\co$. Let
\[
g=\bigcup_{\xi<\co}(b_\xi+f\restriction X_\xi)
\]
and note that, by (\ref{eqFff}), $g(x)=b_\xi+f(x)\neq g_\zeta(x)$ for any
$x\in X_\xi$ and $\zeta<\lambda_\xi$. In particular $g\in \sz$.
Therefore, by (\ref{eqFee}),
\[
\dec(\{f\},\sC)\leq\!\sup_{\xi<\cf(\co)}\!\dec(\{f\!\restriction X_\xi\},\sC)
=\!\sup_{\xi<\cf(\co)}\!\dec(\{b_\xi+f\!\restriction X_\xi\},\sC)\leq
\dec(\{g\},\sC)
\]
and so
\[\kappa\leq\dec(\{g\},\sC)\leq\dec(\sz,\sC).\]
This finishes the proof of Theorem~\ref{th1a}.


\section{Proof of Theorem~\ref{th2}} 

Part (a) of Theorem~\ref{th2} holds in a Cohen model
obtained by adding $\lambda=\co$ Cohen reals.
This follows from the fact, proved by
G.~Gruenhage (see Rec{\l}aw~\cite[Theorem~4]{Recl}) and
S.~Shelah~\cite{Sh2} (see also \cite{KC:STAsurvey}),
that in such a model there exists an
$f\colon\real\to\real$ for which
$f\restriction X$ is discontinuous
for every uncountable $X\subset\real$.

The fact that Theorem~\ref{th2}(a) holds in a Cohen model
can also be easily proved directly. (Some difficulty in the result of
Gruenhage and Shelah comes from the fact that their
function is defined on the entire real line. For our
proof, however, it is enough to have a partial function $f$
defined on a set of cardinality $\co$ with the same property.)
Simply, let
$\{x_{\la\xi,i\ran}\colon\la\xi,i\ran\in\lambda\times 2\}$
be a one-to-one enumeration of the Cohen reals and
define $f$ on $X=\{x_{\la\xi,0\ran}\colon\xi<\lambda\}$
by $f(x_{\la\xi,0\ran})=x_{\la\xi,1\ran}$.
Then $f$ has the desired property.\footnote{It has been pointed by the referee that
the existence of such a partial function $f$ follows from the existence
Lusin set (also Sierpinski set) of size continuum. Simply, choose $f$
of size continuum in the Lusin subset of the plane
of size continuum.
}



In the proof of part (b) of Theorem~\ref{th2}
we will need the following lemma, which is
an easy variation of a result in
Baldwin~\cite{Ba} that under the Martin's Axiom MA for every function
$f\colon\real\to\real$ and every infinite $\kappa<\co$ there exists
a $\kappa$-dense set $X\subset\real$ such that $f\restriction X$ is continuous.


\lem{lem1}{ If MA holds, then for every $X\subset\real$ with
cardinality less than $\co$ and for every $f\colon X\to\real$
there exists a countable partition $\{X_n\colon n<\omega\}$
of $X$ such that $f\restriction X_n$ is continuous for every $n<\omega$.}

\begin{proof}
Let $D$ be a countable dense subset of $[-\infty,\infty]\setminus X$
which contains $\{-\infty,\infty\}$ and let
$\mathcal{S}$ be the family of all finite
unions $\bigcup_{i=0}^n(a_i,b_i)\times(c_i,d_i)$,
where $a_i,b_i,c_i,d_i\in D$,
$a_i<b_i$, $c_i<d_i$, and the intervals $\{(a_i,b_i)\}_{i=0}^n$
form a disjoint cover of $\real\setminus D$.
Moreover, for $k>0$ let $ \mathcal{ S}_k$ be the family
of these unions $\bigcup_{i=0}^n(a_i,b_i)\times(c_i,d_i)$
from $\mathcal{ S}$ for which $(a_i,b_i)$ and $(c_i,d_i)$ have lengths
less than $1/k$ for every $i$ with $(a_i,b_i)\cap(-k,k)\neq\emptyset$.

Consider the forcing
\[
R_f=\{\la A,S\ran\colon A\in[X]^{<\omega}\ \&\ f\restriction A\subset
S\in \mathcal{ S}\}
\]
ordered by $\la A,S\ran\leq\la B,T\ran$
if $B\subset A$ and $S\subset T$. Define $P_f$ as a finite support
product of forcings $R_f$, that is, $P_f$ is the set of all
sequences $\la \la A_j,S_j\ran\colon j<\omega\ran$ from $(R_f)^\omega$
for which $\la A_j,S_j\ran=\la \emptyset,\real^2\ran$ for all but
finitely many $j$'s.

It is easy to see that $P_f$ is ccc (in fact, it is $\sigma$-centered)
since the family $ \mathcal{ S}$ is countable and any
conditions from $R_f$ with the same
second coordinate are compatible.
Next\pagebreak\ notice that the following subsets of $P_f$ are dense
for every $x\in X$ and $i<k<\omega$:
\[
D_x=\{\la A_j,S_j\ran_{j<\omega}\in P_f\colon x\in\bigcup_{j<\omega}A_j\},
\]
\[
E_{i,k}=\{\la A_j,S_j\ran_{j<\omega}\in P_f\colon S_i\in \mathcal{ S}_k\}.
\]
Let $\sG=\{D_x\colon x\in X\}\cup\{E_{i,k}\colon i<k<\omega\}$
and let $\sF$ be a $\sG$-generic filter in $P_f$.
For $i<\omega$ we put
\[
X_i=\bigcup\{A_i\colon \la A_j,S_j\ran_{j<\omega}\in\sF\}.
\]
Then the sets $D_x$ guarantee that $\bigcup_{i<\omega}X_i=X$,
while the sets $E_{i,k}$ force that each restriction
$f\restriction X_i$ is continuous. \end{proof}


Now, to prove Theorem~\ref{th2}(b) start with a model $M$ of ZFC+GCH
and take a cardinal $\lambda$ with uncountable cofinality.

If $\cf(\lambda)=\lambda$, then (b) holds in a model from
 part (a).
Thus we will assume that $\cf(\lambda)<\lambda$.
Let $\{\lambda_\xi\colon\xi<\cf(\lambda)\}$
be an increasing sequence cofinal
with $\lambda$ such that each $\lambda_\xi$ is a cardinal successor.
Define $P$ as a finite support iteration of forcings
$M_\xi$, where each $M_\xi$ is a standard ccc forcing
adding the Martin's Axiom over the previous model and making
$\co=\lambda_\xi$. Let $G$ be an $M$-generic filter over $P$.
We claim that (b) holds in $M[G]$.

Checking that $\co=\lambda$ in $M[G]$ is routine.
To see that $\dec(\real^\real,\sC)=\cf(\lambda)$
it is enough to show that
$\dec(\{f\},\sC)\leq\cf(\lambda)$ for every $f\in\real\to\real$.
So fix $f\in\real^\real$ from $M[G]$ and
let $\hat f$ be a $P$-name for $f$.
For $\xi<\cf(\lambda)$ let
$X_\xi$ be the set of all $x\in\real$
for which the value of $\hat f(x)$ is already decided
in the model $M[G\cap P_\xi]$, where $P_\xi$ is the iteration
of forcings $M_\zeta$ up to $\xi$.
Then $\real=\bigcup_{\xi<\cf(\lambda)}X_\xi$ and, by Lemma~\ref{lem1},
in $M[G\cap P_{\xi+1}]$
there is a cover $\{X_\xi^n\colon n<\omega\}$ of $X_\xi$
such that each function $f\restriction X_\xi^n$ is continuous.
Then functions
$\{f\restriction X_\xi^n\colon \xi<\cf(\lambda)\ \&\ n<\omega\}$
witness $\dec(\{f\},\sC)\leq\cf(\lambda)$
in $M[G]$, finishing the proof.





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\end{document}
