%\documentstyle[12pt]{article}


\documentstyle[amsfonts,amssymb,amsmath,12pt]{article}

\newtheorem{Th}{Theorem}[section]
\newtheorem{Le}{Lemma}[section]
\newtheorem{Co}{Corollary}[section]
\newtheorem{Po}{Proposition}[section]
\newtheorem{Ro}{Remark}[section]
\newtheorem{Pro}{Problem}
\newcommand\mathN{{\mathbb N}}
\newcommand\mathI{{\mathbb I}}
\newcommand\mathR{{\mathbb R}}
\newcommand\mathQ{{\mathbb Q}}
\newcommand\real{\mathR}
\newcommand{\RRR}{\mathR^{\mathR}}
\newcommand{\Qed}{\unskip\nolinebreak\quad\hfill$\Box\;\;$\medskip}


\newcommand{\F}{{\cal F}}
\def\P{{\cal P}}
\def\E{{\cal E}}
\def\L{{\cal L}}
\def\M{{\cal M}}
\def\N{{\cal N}}
\newcommand{\A}{{\cal A}}
\newcommand{\B}{{\cal B}}
\newcommand{\C}{{\cal C}}
\newcommand{\K}{{\cal K}}
\newcommand{\X}{{\cal X}}
\newcommand{\dom}{{\rm dom}\,}
\newcommand{\rng}{{\rm rng}\,}

\newcommand{\G}{{\cal G}}
\newcommand{\I}{{\cal I}}
\newcommand{\CC}{{\rm C}}
\newcommand{\D}{{\rm D}}
\newcommand{\Conn}{{\rm Conn}}
\newcommand{\Ext}{{\rm Ext}}
\newcommand{\ACS}{{\rm ACS}}
\newcommand{\MM}{{\rm M}}

\newcommand\Int{{\rm int}\,}
\newcommand{\pf}{{\noindent\sc Proof. }}
\newcommand{\restr}{{\hbox{$|\grave{}$}}}
\newcommand{\co}{{\frak c}}

\def\rl{{\mathR}}


\def\ds{\bigcup}
\def\di{\bigcap}
\def\sb{\subset}
\def\su{\subseteq}

\def\sm{\setminus}
\def\es{\emptyset}


\title{Universal summands for families of measurable functions}
\author{Tomasz Natkaniec \and Ireneusz Rec\l aw}
\date{(Gda{\'n}sk)}

\begin{document}

{
\renewcommand{\thefootnote}{}
 \footnotetext{AMS Classification: primary 04A15;  secondary
 26A15; 28A05; 28A20.}
}
{
{\renewcommand{\thefootnote}{}
\footnotetext{Key words: Darboux functions, almost continuous
functions, extendable functions,  universal set, universal function,
universal summand.
}
}

 \maketitle
 \begin{abstract}
(1)~For any $\alpha<\omega_1$ there exists a Borel measurable
function $g\colon\mathR\to\mathR$ such that $g+f$ is a Darboux
function (is almost continuous in the sense of Stallings) for every
$f\in B_{\alpha}$. This solves a problem of J.~Ceder. (2)~There is a
function $g$ that is universally measurable and has the Baire
property in restricted sense such that $g+f$ is Darboux for every
Borel measurable function $f$. (3)~There is $g\colon\mathR\to\mathR$
such that $f+g$ is extendable for each $f\colon\mathR\to\mathR$ that
is Lebesgue measurable (has the Baire property). (4)~For every
$\alpha<\omega_1$, each $f\in B_{\alpha}$ is the sum of two
extendable functions $f_1,f_2\in B_{\alpha}$. This answers a question
of A.~Maliszewski.
\end{abstract}


\section{Preliminaries.}
Our terminology is standard.
By $\mathR$ and $\mathI$ we denote the set of all reals and the
interval $[0,1]$, respectively. If $A$ is a planar set,
 we denote its $x$-projection by $\dom(A)$ and $y$-projection by
$\rng(A)$. The symbol $|X|$ stands for the cardinality of a set $X$.
The cardinality of $\mathR$ is denoted by $\co$.
All notions and properties of Borel and projective sets that we use
can be found in~\cite{K}.

For $A\subset X\times Y$ and $x\in X$ we denote by $A_x$ the {\it $x$-
section} of $A$, i.e.,  the set $A_x=\{ y\in Y\colon\, (x,y)\in A\}$.
Similarly, if $f\colon X\times Y\to Z$ and $x\in X$, then $f_x$
denotes the {\it $x$-section} of $f$, i.e., $f_x(y)=f(x,y)$ for each
$y\in Y$.

Let $C$ be a Cantor set (i.e., a non-empty nowhere dense perfect
subset of $\mathR$). We say that $A\subset C\times X$
is a {\it ``$C$-universal'' set} for a class $\P$ of subsets of $X$
iff $\P\subset\{ A_x\colon\, x\in C\}$. Similarly, we say that
$f\colon C\times X\to Y$ is a {\it ``$C$-universal'' function} for a
family $\F$ of functions from $X$ to $Y$ iff $\F\subset\{ f_x\colon
x\in C\}$.

We shall deal mainly with real functions of one real variable.  No
distinction is made between a function and its graph.
By $\CC$ we denote the class of all continuous functions from
$\mathR$ to $\mathR$. For $X=\mathR$ or $X=\mathR^2$ the symbols
$\L(X)$, $\N(X)$, $\K(X)$ and $\M(X)$ stand for the families of
Lebesgue measurable sets, measure zero sets, Baire sets and meager
sets, respectively. Moreover, the symbols $\L(X:\mathR)$ and
$\K(X:\mathR)$ denote the families of all Lebesgue measurable
functions and of all functions with the Baire property from $X$ to
$\mathR$.

We shall consider the following classes of functions from $X$ to $Y$:
\begin{description}
\item[D]
 -- $f$ is a {\it Darboux function} if $f(C)$ is connected
 whenever $C$ is connected in $X$;
\item[Conn]
 -- $f$ is a {\it connectivity function} if the graph of $f$
 restricted to $C$, denoted by $f\restr C$, is connected in
 $X\times Y$ whenever $C\subset X$ is connected;
\item[ACS]
 -- $f$ is an {\it almost continuous function} in the sense of
 Stallings, if $U$ is an open subset of $X\times Y$ containing
 the graph of $f$, then $U$ contains the graph of a continuous
 function $g\colon X\to Y$~\cite{JS};
\item[Ext]
 -- $f$ is an {\it extendable function} if there exists a
 connectivity function $g\colon X\times
\mathI\to Y$ such that $f(x)=g(x,0)$ for all $x\in X$~\cite{JS}.
\end{description}

Recall that for $X=Y=\mathR$ we have the following chain of proper
inclusions
$$\CC\subset\Ext\subset\ACS\subset\Conn\subset\D\subset\RRR$$

For a given family $\F$ of real functions we can examine the
following question:
\begin{quote}
{\em
For which families $F\subset\RRR$ does there exist a {\it $\F$-
universal summand}, i.e., such $g\in\RRR$ that $f+g\in\F$ for all
$f\in F$?
}
\end{quote}
This question was considered by many authors. The first result in
this direction was obtained by H.~Fast in 1959.
\begin{Th}{\rm \cite{HF}}
If $F$ is a family of functions and $|F|\le \co$, then there exist a
$\D$-universal summand for $F$.
\end{Th}
The analogous theorem for the class of all almost continuous
functions was obtained in 1974 by K.~Kellum~\cite{KK}.
Those results were generalized in 1994 by K.~Ciesielski and A.~Miller
in~\cite{CMi} and, for extendable functions, in 1995 by K.~Ciesielski
and I.~Rec{\l}aw~\cite{CR}.

$\D$-universal summands for families of Borel measurable functions
were studied by J.~Ceder in~\cite{JC}. He proved that for every
$\alpha<\omega_1$ and for any countable family $F\subset B_{\alpha}$
there exists a $D$-universal summand for $F$ that is Borel
measurable. Moreover, he asked whether the analogous theorem is valid
for every $F\subset B_{\alpha}$ with $|F|\le\co$, i.e., whether there
is a Borel measurable $\D$-summand for the class $B_{\alpha}$. We
shall answer this question in the affirmative. Moreover, we shall
prove the analogous theorem for $\ACS$-summands.

By Fast's Theorem, there exist $\D$-universal summands for the family
$B$ of all Borel measurable functions. It is easy to observe that
every such summand does not belong to $B$. We shall show that such a
summand can be universally measurable and with the Baire property in
restricted sense.

Finally, it is easy to observe that there is no $\D$-universal
summand for the family $\RRR$. On the other hand, there are $\D$-
universal summands for big ``regular'' families of functions. In
particular, there are $\ACS$-universal summands for the family
$\L(\mathR:\mathR)$ of all Lebesgue measurable functions and for the
family $\K(\mathR:\mathR)$ of all functions with the Baire
property~\cite{TN}. We shall prove that there are $\Ext$-universal
summands for $\L(\mathR:\mathR)$ and $\K(\mathR:\mathR)$.

\section{D-universal summands.}
Let $A(X)$ be one of the classes: $\Sigma_\alpha^0$ for $\alpha <
\omega_1$ or $\Sigma_n^1$ for $n<\omega$.
Then let $\underline{\MM}A(X)=\{f\colon X \to [0,1]\colon\, (\forall
c\in [0,1))\,\, f^{-1}((c,1]) \in A(X)\}$.
J.~Cicho{\'n} and M.~Morayne proved the following theorem:
\begin{Th}{\rm~\cite{CM}}
There is $ f\in \underline{\MM}A(C\times C)$ that is ``$C$-
universal'' for the class $\underline{\MM}A(C)$.
\Qed
\end{Th}
From this theorem we can easily deduce the following two lemmas.
\begin{Le}\label{l1}
Let $C$ be a Cantor set. For each $\alpha < \omega_1$ there is a
function $f\colon C\times \rl \to \rl$ of the Baire class $\alpha +2$
that is ``$C$-universal'' for the class $B_{\alpha}$.\Qed
\end{Le}

\begin{Le}\label{l2}
Let $C$ be a Cantor set. For each $n\in\omega$ there is a
$\sigma(\Sigma_n^1)$-measurable function $f\colon C\times \mathR\to
\rl$ that is ``$C$-universal'' for the class of all $\Sigma_n^1$-
measurable functions.\Qed
\end{Le}

\begin{Th}\label{t1}
For every $\alpha < \omega_1$ there is a function $h\in B_{\alpha+2}$
such that for each  function
$g\in B_{\alpha}$ the set $\{x\in \rl\colon h(x)=g(x)\}$
is dense in $\mathR$.
\end{Th}
\pf
Let $\{U_n\colon n\in\omega\}$ be a basis of $\mathR$. Let $\{
P_n\colon n\in\omega\}$ be a family of pairwise disjoint perfect sets
with $P_n \sb U_n$ for each $n$. Fix $n\in\omega$. By Lemma~\ref{l1},
there exists $f_n\colon P_n\times\mathR\to\mathR$ that is ``$P_n$-
universal'' for the class $B_{\alpha}$ and that belongs to the class
$B_{\alpha+2}$. Let $h_n\colon P_n \to \rl$ be the diagonal of $f_n$,
i.e., $h_n(x)=f_n(x,x)$ for $x\in P_n$. Define $h\colon\mathR\to\rl$
by
$$h(x)=\left\{ \begin{array}{ll}
h_n(x) &\mbox{for $x \in P_n$,}\\
0 & \mbox{otherwise.}
\end{array}\right.$$
It is easy to observe that $h\in B_{\alpha+2}$. Moreover, for each $n
\in
\omega$ and $g\in B_{\alpha}$ there is $x \in P_n \sb U_n$ with
$f_n(x,x)=g(x)$, so $h(x)=g(x)$.\Qed

Analogously we can prove the following theorem.
\begin{Th}\label{t2}
For each $n\in\omega$ there is a
$\sigma(\Sigma_n^1)$-measurable function
$h\colon\rl\to\rl$  such that for every  $\Sigma_n^1$-measurable
$g\colon\rl\to\rl$  the set $\{x\in \rl\colon h(x)=g(x)\}$ is dense
in $\mathR$.
\Qed
\end{Th}

\begin{Co}\label{c1}
For each $\alpha<\omega_1$ there is $k\in B_{\alpha+2}$ that is a
$\D$-universal summand for the class $B_{\alpha}$.
\end{Co}
\pf
Put simply $k= -h$, where $h$ is defined in Theorem~\ref{t1}. Then
$k$ is a $\D$-universal summand for the class $B_{\alpha}$. Indeed,
fix $g\in B_{\alpha}$.
Then for each $r \in \rl$, $g-r\in B_{\alpha}$. Thus
$\{x\in \rl\colon r=g(x)+k(x)\}=\{x\in \rl\colon h(x)=g(x)-r\}$ is
dense in $\mathR$, so $f+g$ is Darboux. \Qed

Observe that if $k$ is $\D$-universal summand for the class
$B_{\alpha}$, $\alpha>0$, then $k\not\in B_{\alpha}$. Indeed, if
there were $k\in B_{\alpha}$ such that $f+g$ were Darboux for each
$f\in B_{\alpha}$, then the function $h\in\RRR$ defined by $h(x)=-
k(x)$ for $x\ne 0$ and $h(0)=-k(0)+1$ belongs to $B_{\alpha}$, but
$k+h$ fails to be Darboux. We are unable to determine whether a $\D$-
universal summand for the class $B_{\alpha}$ can be found in the
class $B_{\alpha+1}$.

Theorem~\ref{t2} yields the following result.
\begin{Co}\label{c2}
For each $n\in\omega$ there is a $\D$-universal summand for the class
of $\Sigma_n^1$-measurable functions that is
$\sigma(\Sigma_n^1)$-measurable.\Qed
\end{Co}

Recall that every $\sigma(\Sigma_1^1)$-measurable function is
universally measurable and has the Baire property in restricted sense
(see, e.g.,~\cite[Theorem 21.10, p.~156]{K}) and every Borel
measurable function is $\Sigma_1^1$-measurable.
Thus we obtain the following corollary.
\begin{Co}
There is a universally measurable and with the Baire
property in restricted sense
 function $k\colon\rl \to \rl$ such that for each Borel function
$g\colon\rl \to \rl$, $k+g$ is Darboux.\Qed
\end{Co}

\section{ACS-universal summands.}
Recall that if $f$ intersects all
 closed subsets $K$ of $ \mathR^2$ with $\dom(K)$ being a
 non-degenerate interval and $\rng(K)=\mathR$, then $f$ is almost
continuous \cite{KK}. In this paper every such set is called a {\em
blocking set}.

\begin{Th}\label{ac1}
For each $\alpha < \omega_1$ there is a Borel function
$k\colon\rl \to \rl$ that is a $\ACS$-universal summand for the class
$B_{\alpha}$.
\end{Th}
\pf
Let $\{U_n\colon n \in \omega\}$ be a basis of $\mathR$ and let
$\{P_n\colon n\in \omega\}$ be a family of pairwise disjoint perfect
sets with $P_n \sb U_n$ for each $n$.
Let $C$ be the ternary Cantor set and let $w_n\colon P_n \to C\times
C$ be a homeomorphism.
Define $j\colon (C\times C) \times \rl \to \rl$ by
$j(c_1,c_2,c_3)=f(c_1,c_3)$, where $f$ is a Borel measurable ``$C$-
universal'' function for the class $B_{\alpha}$. (See
Lemma~\ref{l1}.) Moreover, let $D_n' \sb C \times (\rl\times \rl)$ be
a closed ``$C$-universal'' set for
closed sets in $\rl\times \rl$. (See e.g.,~\cite[Theorem 22.3,
p.~168]{K}.)
Define $D_n \sb (C \times C) \times (\rl\times \rl)$ by the formula
$$(c_1,c_2,c_3,c_4) \in D_n \equiv (c_2,c_3,c_4) \in D_n'.$$
For every $n\in\omega$ define the function $k_n\colon P_n
\times \rl \to \rl$ by
$k_n(p,x)=j(w_n(p),x)$ and the set $E_n=\{(p,x,y)\colon
(w_n(p),x,y)\in D_n\}$.
Observe that $k_n$ and $E_n$ have the following properties:
\begin{itemize}
\item
$k_n$ is Borel measurable (in fact, $k_n\in B_{\alpha+2}$);
\item
 $E_n$ is closed;
 \item
for each pair $(g,F)$, where $g\colon\rl\to\rl$, $g\in B_{\alpha}$
and $F$ is a closed subset of $\rl\times\rl$,
there is $p\in P_n$ such that $(E_n)_p=F$ and
$(k_n)_p=g$.
\end{itemize}
Then all sets
$$W_n=\{(p,y)\in P_n\times \rl\colon\, y \in ((E_n)_p)_p-k_n(p,p)\}$$
are Borel measurable. Indeed, it is easy to observe that for each
$n\in\omega$ the set $E'_n=\{ (p,y)\in P_n\times\mathR\colon\,
(p,p,y)\in E_n,\, p\in P_n\}$ is closed. Moreover, the function
$\varphi_n\colon P_n\times \mathR\to P_n\times\mathR$ defined by
$\varphi_n(p,y)=(p,y-k_n(p,p))$ is injective and Borel measurable.
Thus, by Lusin-Souslin's Theorem (\cite[Theorem~15.1, p.~89]{K}),
$W_n=\varphi_n(E'_n)$ is a Borel set. Additionally, all sections of
$W_n$ are closed, so $\sigma$-compact.
Thus Arsenin-Kunugui's Theorem (\cite[Theorem~18.18, p.~127.]{K})
follows that $W_n$ has a Borel
uniformization, i.e., there exists $u_n\colon\dom(W_n) \to \rl$ with
$u_n\sb W_n$. Let $u\colon\rl \to \rl$ be any Borel extension of all
$u_n$, $n\in\omega$.

We shall show that $u$ is an $\ACS$-universal summand for the class
$B_{\alpha}$. So, fix $f\in B_{\alpha}$ and a blocking set $F\subset
\mathR^2$.
 Then there are $n\in\omega$ and $p\in P_n$ such that $P_n \sb
\dom(F)$, $(k_n)_p=f$ and $(E_n)_p=F$. In
particular,
$k_n(p,p)=g(p)$ and $((E_n)_p)_p)=F_p$. Since $F_p$ is non-empty,
$u(p)\in F_p - g(p)$. Consequently,  $(p,u(p)+g(p))\in F$. Thus $u+g$
intersects each blocking sets, so it is almost continuous.\Qed

\section{Ext-universal summands.}
\begin{Le}\label{ext-l}
There exists a family $\E$ of pairwise disjoint perfect sets such
that $|\{ E\in\E\colon\; E\subset A\}|=\co$ for each
$A\in\L(\mathR)\setminus\N(\mathR)$.
\end{Le}
\pf
Let $\varphi\colon\mathR\to\mathR^2$ be a Borel isomorphism that maps
the ideal $\N(\mathR)$ onto $\N(\mathR^2)$. (See,
e.g.,~\cite[Theorem~17.41. p.~116]{K}.) Let $\{ G_{\alpha}\colon\,
\alpha<\co\}$ be the family of all Borel subsets of $\mathR$ that
does not belong to $\N(\mathR)$. For every $\alpha<\co$,
$\varphi(G_{\alpha})\not\in\N(\mathR^2)$, so $A_{\alpha}=\{ x\colon\,
|(\varphi(G_{\alpha}))_x|=\co\}$ is of the size $\co$. For each
$\alpha$ choose $x_{\alpha}\in A_{\alpha}$ such that $x_{\alpha}\ne
x_{\beta}$ for $\alpha\ne \beta$, and put $B_{\alpha}=\{
x_{\alpha}\}\times (\varphi(G_{\alpha}))_{x_{\alpha}}$. Since
$\varphi^{-1}(B_{\alpha})$ is a Borel set in $\mathR$ and $|\varphi^{-
1}(B_{\alpha})|=\co$, there is a perfect set $C_{\alpha}\subset
\varphi^{-1}(B_{\alpha})\subset G_{\alpha}$. Note that
$C_{\alpha}\cap C_{\beta}\subset\varphi^{-1}( B_{\alpha} \cap
B_{\beta})=\emptyset$ whenever $\alpha\neq \beta$. Finally decompose
each $C_{\alpha}$ onto $\co$ many perfect sets $E_{\alpha,\beta}$,
$\beta<\co$, and put $\E=\{ E_{\alpha,\beta}\colon\,
\alpha,\beta<\co\}$. \Qed

\begin{Th}\label{ext1}
There exists an $\Ext$-universal summand for the family
$\L(\mathR:\mathR)$.
\end{Th}
\pf
Let $B=\{ f_{\beta}\colon \beta<\co\}$ be the family of all Borel
measurable functions from $\mathR$ to $\mathR$ and let $\{
G_{\alpha}\colon \alpha<\co\}$ be the family of all Borel sets
$G\subset\mathR$ that $\mathR\setminus G\in \N(\mathR)$. Applying
Lemma~\ref{ext-l} choose a family of pairwise disjoint $F_{\sigma}$ c-
dense sets $E_{\alpha}\subset G_{\alpha}$, $\alpha<\co$. Next divide
each $E_{\alpha}$ onto $\co$ many $F_{\sigma}$ c-dense sets
$E_{\alpha,\beta}$, $\beta<\co$. By \cite[Lemma 3.2]{CR}, for each
pair $(\alpha,\beta)\in\co\times\co$ there exists
$g_{\alpha,\beta}\in\Ext$ such that $\mathR\setminus
E_{\alpha,\beta}$ is $g_{\alpha,\beta}$-negligible, i.e., every
$f\colon\mathR\to\mathR$ with $f\restr
E_{\alpha,\beta}=g_{\alpha,\beta}\restr E_{\alpha,\beta}$ is an
extendable function. Define $g\colon\mathR\to\mathR$ by
$$g(x)=\left\{ \begin{array}{ll}
g_{\alpha,\beta}(x)-f_{\beta}(x) &\mbox{for $x\in E_{\alpha,\beta}$,
$\alpha,\beta<\co$,}\\
0&\mbox{otherwise.}
\end{array}\right.$$
We shall verify that $g$ is an $\Ext$-universal summand for the class
$\L(\mathR:\mathR)$. For fixed $f\in\L(\mathR:\mathR)$ choose
$\alpha,\beta<\co$ such that $A=\{ x\in\mathR\colon f(x)\ne
f_{\beta}(x)\}\in\N(\mathR)$ and $G_{\alpha}\subset\mathR \setminus
A$. Then $f(x)+g(x)=g_{\alpha,\beta}(x)$ for $x\in E_{\alpha,\beta}$,
and therefore $f+g$ is an extendable function.\Qed

\noindent
{\bf Remark. }Note that an $\Ext$-universal summand for the family
$\L(\mathR:\mathR)$ cannot be measurable.

Analogously we can prove the following result
\begin{Th}\label{ext2}
There exists an $\Ext$-universal summand for the family of all
functions possessing the Baire property.\Qed
\end{Th}

\begin{Pro}
Let $0<\alpha<\omega_1$. Does there exist a Borel measurable $\Ext$-
universal summand for the class $B_{\alpha}$?
\end{Pro}

Finally, note that Fast's Theorem implies an old result of Lindenbaum.
\begin{Th}{\rm \cite{AL}}
Every function from $\mathR$ to $\mathR$ can be expressed as the sum
of two Darboux functions.
\end{Th}
Indeed, let $g$ be a $\D$-universal summand for the family $\{ f,
0\}$. Then $g=g+0$ and $g+f$ are Darboux, and $f=(g+f)-g$.
Similarly, Ceder's Theorem implies that every Borel measurable
functions can be expressed as the sum of two Borel measurable Darboux
functions, and, by Kellum's Theorem, every  functions is the sum of
two almost continuous functions. Note that Theorem~\ref{ac1} implies
that every Borel measurable function can be written as the sum of two
Borel measurable almost continuous functions. This solves Problem~1.3
from~\cite{AM}. However, applying \cite[Theorem 3.3]{CR}, we can
easily obtain a more exact  result.
\begin{Th}
For every $\alpha<\omega_1$ and for each $f\in B_{\alpha}$ there are
two extendable functions $f_1,f_2\in B_{\alpha}$ with $f=f_1+f_2$.
Similarly, every $f\in\L(\mathR:\mathR)$ ($f\in\K(\mathR:\mathR)$)
can be written as the sum of two extendable functions
$f_1,f_2\in\L(\mathR:\mathR)$ ($f_1,f_2\in\K(\mathR:\mathR)$).
\end{Th}
\pf
Fix $f\in B_{\alpha}$. If $\alpha=1$, then $f$ is the sum of two
Darboux functions $f_1,f_2\in B_1$~\cite{BCK}, and since $\Ext\cap
B_1=\D\cap B_1$~\cite{BHL}, $f_1$ and $f_2$ are extendable. So assume
that $\alpha\ge 2$. Let $\hat{f}\colon\mathR\to\mathR$ be the
function constructed in \cite[Theorem 3.3]{CR}. It is easy to observe
that $\hat{f}\in B_2$. Since $\hat{f}$ is dense in $\mathR^2$, there
exists an $F_{\sigma}$ set $E\subset\mathR$ such that every
$g\colon\mathR\to\mathR$ with $\hat{f}\restr E=g\restr E$ is
extendable. (See~\cite{HR}.) Let $h\colon\mathR\to\mathR$ be a
homeomorphism such that $h(E)\cap E=\emptyset$. (See~\cite{WG}.) Then
$\hat{g}=\hat{f}\circ h^{-1}$ is an extendable function of the Baire
class two and every $g\colon\mathR\to\mathR$ with $g\restr
h(E)=\hat{g}\restr h(E)$ is an extendable function. (See~\cite{TN1}.)
Consequently the functions
$$\begin{array}{cc}
f_1(x)=\left\{\!\! \begin{array}{ll}
\hat{f}(x) &\!\!\!\mbox{for $x\in E$,}\\
f(x)\!-\!\hat{g}(x) &\!\!\!\mbox{for $x\in h(E)$,}\\
f(x) &\!\!\!\mbox{otherwise}
\end{array}\right.
&\!\!\!
f_2(x)=\left\{\!\! \begin{array}{ll}
f(x)\!-\!\hat{f}(x) &\!\!\!\mbox{for $x\in E$,}\\
\hat{g}(x) &\!\!\!\mbox{for $x\in h(E)$,}\\
0 &\!\!\!\mbox{otherwise}
\end{array}\right.
\end{array}$$
are extendable and belong to the class $B_{\alpha}$, and $f=f_1+f_2$.
\Qed

Note that in the analogous way we can prove that for every countable
family of $B_{\alpha}$ functions there exists an $\Ext$-universal
summand in the class~$B_{\alpha}$.

\medskip
\begin{center}
\sc
Department of Mathematics, Gda\'{n}sk University\\
Wita Stwosza 57, 80-952 Gda{\'n}sk, Poland\\
(E-mail: mattn@ksinet.univ.gda.pl, \, matir@paula.univ.gda.pl)
\end{center}

\begin{thebibliography}{123}
\bibitem[BHL]{BHL}
J.~Brown, P.~Humke and M.~Laczkovich, {\em Measurable Darboux
functions}, Proc. Amer. Math. Soc. {\bf 102} (1988), 603--610.
\bibitem[BCK]{BCK}
A.~M.~Bruckner, J.~Ceder and R.~Keston, {\em Representations and
approximations by Darboux functions in the first class of Baire},
Rev. Roum. Math. Pures et Appl. {\bf 13} (1968), 1247--1254.
\bibitem[JC]{JC}
J.~Ceder, {\em On representing functions by Darboux functions}, Acta
Sci. Math. (Szeged) {\bf 26} (1965), 283--288.
\bibitem[CM]{CM}
 J.~Cicho{\'n} and M.~Morayne, {\em Universal functions and
generalized classes of functions}, Proc. of Amer. Math. Soc. {\bf
102} (1988), 83--89.
\bibitem[CMi]{CMi}
 K.~Ciesielski and A.~W.~Miller, {\em Cardinal invariants
 concerning functions whose sum is almost continuous}, Real Anal.
Exchange {\bf 20} (1994--95), 657--672.
\bibitem[CR]{CR}
  K.~Ciesielski and I.~Rec{\l}aw, {\em Cardinal invariants
 concerning extendable and peripherally continuous continuous
functions}, Real Anal. Exchange {\bf 21} (1995--96), 459--472.
\bibitem[HF]{HF}
 H.~Fast, {\em Une remarque sur la propri{\'e}t{\'e} de
 Weierstrass}, Colloq. Math. {\bf 7} (1959), 75--77.
\bibitem[WG]{WG}
 W.~Gorman~III, {\em The homeomorphic transformation of c-sets into d-
sets}, Proc. Amer. Math. Soc. {\bf 17} (1966), 825--830.
\bibitem[AK]{K}
A.~S.~Kechris, {\em Lectures on classical descriptive set
theory}, Springer Verlag, Berlin 1995.
\bibitem[KK]{KK}
 K.~R.~Kellum, {\em Sums and limits of almost continuous
 functions}, Colloq. Math. {\bf 31} (1974), 125--128.
\bibitem[AL]{AL}
A.~Lindenbaum, {\em Sur quelques propri{\'e}ti{\'e}s des
 fonctions de variable r{\'e}ele}, Ann. Soc. Math. Polon {\bf 6}
 (1927), 129--130.
 \bibitem[AM]{AM}
 A.~Maliszewski, {\em Darboux property and quasi-continuity. A
uniform approach}, S{\l}upsk 1996.
\bibitem[TN]{TN}
 T.~Natkaniec, {\em Almost continuity}, Real Anal. Exchange {\bf 17}
(1991--92), 462--520.
\bibitem[TN1]{TN1}
 T.~Natkaniec, {\em Extendability and almost continuity},
 Real Anal. Exchange {\bf 21} (1995--96), 349--355.
\bibitem[HR]{HR}
 H.~Rosen, {\em Limits and sums of extendable
 connectivity functions}, Real Anal. Exchange {\bf 20} (1994--95),
183--191.
\bibitem[JS]{JS}
 J.~Stallings, {\em Fixed point theorems for connectivity maps},
 Fund. Math. {\bf 47} (1959), 249--263.
\end{thebibliography}
\end{document}