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\begin{document}
%\baselineskip=23pt

\title{Sum of Sierpi{\'n}ski-Zygmund and Darboux Like Functions}

\author{Krzysztof P\l otka\thanks{This paper was written under supervision of K. Ciesielski.
The author wishes to thank him for many helpful conversations.}\\
	Department of Mathematics, West Virginia University\\
       Morgantown, WV 26506-6310, USA\\ 
       kplotka@math.wvu.edu \\
                 and \\
        Institute of Mathematics, Gda{\'n}sk University\\
        Wita Stwosza 57, 80-952 Gda{\'n}sk, Poland}

%\date
\maketitle

\begin{abstract}
For ${\cal F}_1, {\cal F}_2 \sq \real^\real$ we define
$\add({\cal F}_1,{\cal F}_2)$ as the smallest cardinality of a family 
$F \sq \real^\real$ for which there is no $g \in {\cal F}_1$ such that
$g+F \sq {\cal F}_2$. The main goal of this note is to investigate the function $\add$ 
in the case when one of the classes ${\cal F}_1, \; {\cal F}_2$ is the class $\sz$ of 
{\it Sierpi{\'n}ski-Zygmund\/} functions. In particular, we show that 
{\it Martin's Axiom \/} (MA)  implies 
$ \add(\ac,\sz) \ge \omega $ and $\add(\sz,\ac)= \add(\sz,\da) = \cont$,
where $\ac$ and $\da$ denote the families of {\it almost continuous \/} and 
{\it Darboux\/} functions, respectively. 
As a corollary we obtain that the proposition: 
{\emph {every function from $\real$ into $\real$ can be represented as a sum of 
Sierpi{\'n}ski-Zygmund and almost continuous functions}} is independent of ZFC axioms.
\end{abstract}

\section{Introduction} \label{S:Intro} 

The terminology is standard and follows \cite{cie}. The symbols $\real$ and $\rational$ 
stand for the sets of all real and all rational numbers, respectively. A basis of $\real$ as a linear space over 
$\rational$ is called {\it Hamel basis\/}.
For $Y\subset\real$, the symbol $\lin(Y)$ stands for the smallest linear subspace of $\real$ over $\rational$
that contains $Y$. The cardinality of a set $X$ we denote by $|X|$. In particular,
$|\real|$ is denoted by $\cont$. Given a cardinal $\kappa$,
we let $\cf(\kappa)$ denote the cofinality of $\kappa$. We say that a cardinal $\kappa$ is regular provided that $\cf(\kappa)=\kappa$.
 
${\cal B}$ and ${\cal M}$ stand for the families
of all Borel and all meager subsets of $\real$, respectively. We say that a set 
$B \sq \real$ is a {\it {Bernstein}} set if both $B$ and $\real \setminus B$ intersect
every perfect set. For a cardinal number $\kappa$, 
a set $A \sq \real$ is called $\kappa${\it -dense\/} if $|A \cap I|\ge \kappa$ 
for every non-trivial interval $I$. For any planar set $P$, we denote its $x$-projection by $\dom(P)$. 

We consider only real-valued functions. No distinction is made between 
a function and its graph. For any two partial real functions $f,g$ we write $f+g$,
$f-g$ for the sum and difference functions defined on $\dom(f) \cap \dom(g)$. 
The class of all functions from a set $X$ into a set $Y$ is denoted by $Y^X$.
We write $f|A$ for the restriction of $f \in Y^X$ to the set $A \sq X$.
For $B\sq\real^n$ its characteristic function is denoted by $\charf{B}$.
If $f, g \in Y^X$, we denote the set $\{ x \in X : f(x)=g(x) \}$ by $[f=g]$. 
For any function $g \in \real^X $ and any family of functions 
$F \sq \real^X$ we define $g+F =\{g+f \colon f\in F\}$.

The cardinal function $\A({\cal F})$, for ${\cal F} \sq \real^X$, is defined 
as the smallest cardinality of a family $F \sq \real^X$ for which 
there is no $g \in \real^X$ such that
$g+F \sq {\cal F}$. It was investigated for many different classes of
real functions, see e.g. \cite{cie-nat}, \cite{cie-rec}, \cite{natkaniec}. 
In this paper we generalize the function
$\A$ by imposing some restrictions on the function g. 
Thus for ${\cal F}_1, {\cal F}_2 \sq \real^X$ we define
$$\add({\cal F}_1,{\cal F}_2) 
= \min \; \{|F|\colon F\sq\real^X \; \& \;  \neg\exists
g\in {\cal F}_1 \; g+F \sq {\cal F}_2 \} \cup\{(|\real^X|)^+\}. $$ 

Observe that  $\A({\cal F})=\add(\real^X,{\cal F})$ for any set $X$, so
the function $\add$ is indeed a generalization of the function $\A$. 
Notice also the following properties of the $\add$ function.

\prop{prop1}{Let ${\cal F}_1 \sq {\cal F}_2 \sq \real^X$ and 
${\cal F} \sq \real^X$. 

\begin{description}

\item[(1)] $\add({\cal F}_1,{\cal F}) \le \add({\cal F}_2,{\cal F})$.

\item[(2)] $\add({\cal F},{\cal F}_1) \le \add({\cal F},{\cal F}_2)$.

\item[(3)] $\add({\cal F}_1,{\cal F}_2) \ge 2$ if and only if 
$\real^X = {\cal F}_2 - {\cal F}_1$.

\item[(4)] If $\add({\cal F}_1,{\cal F}_2) \ge 2$ then 
${\cal F}_1 \cap {\cal F}_2 \not= \e$.

\item[(5)] $\A({\cal F})=\add({\cal F},{\cal F})+1$. In particular, if 
$\A({\cal F}) \ge \omega$ then $\add({\cal F},{\cal F})=\A({\cal F})$.\footnote{Very similar observation, in a little bit different context, was obtained independently by Francis Jordan~\cite[Proposition 1.3]{jordan}.}

\end{description}

}

\proof The properties (1)-(4) are obvious. We will prove (5). 
It is clear that $\add({\cal F},{\cal F}) \le \A({\cal F})$. 
On the other hand, observe that $\A({\cal F}) \le \add({\cal F},{\cal F})+1$.
To see the above let $F \sq \real^\real$ be  such that 
$|F|=\add({\cal F},{\cal F})$ and 
$$\neg \exists \; g \in {\cal F} \;\; g+F \sq {\cal F}.$$
Then we have 
$$\neg \exists \; g \in \real^\real \;\; g+(F\cup \{{\bf 0}\}) \sq {\cal F},$$ where ${\bf 0} \colon \real \to \real$ is a function identically equal to zero.

So the conclusion is obvious in the case $\A({\cal F}) \ge \omega$. Therefore we will concentrate on the case $\A({\cal F})=k$ for some $k \in \omega$. 
Recall that the function $\A$ is bounded from the bottom by $1$, thus $k \ge 1$. 
From the previous argument we imply that $\add({\cal F},{\cal F}) \ge k-1$. So we only need to 
justify that $\add({\cal F},{\cal F}) \le k-1$.

Let $\{f_1, \dots, f_k\}$ be a family witnessing $\A({\cal F})=k$. Then 
the set $\{f_1-f_k, \dots, f_{k-1}-f_k\}$ witnesses $\add({\cal F},{\cal F}) \le k-1$. 
Indeed, assume by contradiction, that we can find a function $f \in
{\cal F}$ such that $(f_i-f_k) + f \in {\cal F}$ for every $i=1, \dots, k-1$. Then the function $f-f_k$ shifts the set $\{f_1, \dots, f_k\}$ into
${\cal F}$. Contradiction.  
\qed



Our main goal is to investigate the function $\add$ in the case when one of
the classes ${\cal F}_1, \; {\cal F}_2$ is the class of 
{\it Sierpi{\'n}ski-Zygmund\/} functions. Before we state the main 
result of the paper, let us recall the following definitions. 

For $X\subseteq\real^n$ a function $f\colon X\to\real$ is:

\begin{itemize}


\item {\em additive\/} if $f(x+y)=f(x)+f(y)$ for all $x,y \in X$
such that $x+y \in X$;

\item {\em almost continuous\/} (in sense of Stallings)  
if each open subset of $X\times\real$ containing the graph of
$f$ contains also graph of a continuous function from $X$ to $\real$;

\item 
 {\it connectivity\/} if the graph of $f|Z$
 is connected in $Z\times\real$ for any connected subset $Z$ of~$X$;

\item 
 {\it countably continuous\/} if it can be represented as a union 
of countably many continuous partial functions;

\item {\em Darboux\/ } if $f[K]$ is a connected subset of $\real$ (i.e.,
an interval) for every connected subset $K$ of $X$;

\item 
 an {\it extendability\/} function provided 
 there exists a connectivity function $F\colon X\times[0,1]\to\real$
 such that $f(x)=F(x,0)$ for every $x\in X$;
 
 \item
{\it peripherally continuous\/} if for
every $x\in X$ and for all pairs of open sets $U$ and $V$ containing
$x$ and $f(x)$, respectively, there exists an open subset $W$ of $U$ 
such that $x \in W$ and $f[\bd (W)]\subset V$;

 \item
{\it Sierpi{\'n}ski-Zygmund\/} if for every set $Y \sq X$ of cardinality
continuum $\cont$, $f|Y$ is discontinuous.

\end{itemize}

The classes of functions defined above are denoted by $\ad(X)$, $\ac(X)$, $\conn(X)$, $\ccf(X)$, $\da(X)$, $\ext(X)$, $\pc(X)$, and $\sz(X)$, respectively. 
The family of all continuous functions from $X$ into $\real$ is denoted by
$\C(X)$. We drop the index $X$ in the case $X=\real $. To simplify
notation, we introduce the symbols $\SZ$ and  $\CC$ to denote 
$ \bigcup_{X \sq \real} \sz(X) $  and   $\bigcup_{X \sq \real} \ccf(X) $.

Recall that a function $f\colon \real^n \to \real$ is almost continuous if and only if it intersects every {\em blocking set\/}, i.e.,
a closed set $K \sq \real^{n+1}$ which meets every continuous function from
$\C(\real^n)$ and is disjoint with at least one function from $\real^{\real^n}$. The domain of every blocking set contains a non-degenerate connected set. (See \cite{kg}.) 
It is also well-known that each continuous
partial function can be extended to a continuous function defined on some
$G_{\delta}$ set. (See \cite{kk}.) Thus if $|[f=g]|< \cont$ for each continuous 
partial function $g$ defined on some $G_{\delta}$-set then $f$ is 
Sierpi{\'n}ski-Zygmund. Recall also that each additive function $f \in \ad$ is linear over $\rational$, i.e., for all $p,q \in \rational$ and 
$x,y \in \real$ we have $f(px+qy)=pf(x)+qf(y)$.

The above classes are related in the following way (arrows
$\longrightarrow$ indicate proper inclusions.) (See \cite{cie-jas} or
\cite{gib-nat}.)


%\vbox{
%\bigskip}


\begin{picture}(0,35)

\put(0,10){\makebox(0,0){$\C$}}
\put(56,10){\makebox(0,0){$\ext$}}
\put(112,10){\makebox(0,0){$\ac$}}
\put(168,10){\makebox(0,0){$\conn$}}
\put(224,10){\makebox(0,0){$\da$} }
\put(280,10){\makebox(0,0){$\pc$}}

\put(10,10){\vector(1,0){30}}
\put(68,10){\vector(1,0){30}}
\put(120,10){\vector(1,0){30}}
\put(185,10){\vector(1,0){30}}
\put(235,10){\vector(1,0){30}}
\end{picture}
\begin{center} For functions from $\real$ into $\real$. \end{center}
			       
%\vbox{
%\bigskip}

\begin{picture}(0,40)
 %\put(0,40){\makebox(0,0){$n \ge 2$}}
 \put(0,20){\makebox(0,0){$\C(\real^n)$}}
 \put(17,20){\vector(1,0){15}}
 \put(110,20){\makebox(0,0){$\ext(\real^n)=\conn(\real^n)=\pc(\real^n)$}}
   \put(184,20){\vector(1,0){15}}
   \put(242,20){\makebox(0,0){$ \ac(\real^n) \cap \da(\real^n)$}}
 \put(283,23){\vector(2,1){14}}
 \put(315,35){\makebox(0,0){${\ac(\real^n)}$}}
 \put(315,5){\makebox(0,0){${\da(\real^n)}$}}
 \put(283,17){\vector(2,-1){14}}
\end{picture}
\begin{center} For functions from $\real^n$ into $\real$ with $n \ge 2$. \end{center}


The class of {\it Sierpi{\'n}ski-Zygmund\/} functions is independent of 
all the classes included in the above chart in the following sense. 
There is no inclusion between $\sz$ and $\ac, \conn, \da,$ or $\pc$.
$\sz$ is disjoint with $\C$ and $\ext$. (See also comment below Corollary 
\ref{consistent}.) $\sz(\real^n)$ is disjoint with 
$\da(\real^n)$ and $\ac(\real^n)$ for  $n \ge 2$. (See Remarks~\ref{ac*sz=empty} and ~\ref{da*sz}.)

The class of additive functions $\ad(\real^n)$ intersects each of the other
classes (the non-emptiness of $\ad \cap \sz$ follows from
Theorem~\ref{other_add2} (iv) and Proposition~\ref{prop1}~(4).)
However, it is not contained in any of them except the family $\pc(\real^n)$
in the case $n=1$. Then we have $\ad \sq \pc$.

Now let us comment on $\A({\cal F})$ for ${\cal F} \in 
\{\ext, \ac, \conn, \da, \pc, \sz\}$. The following can be proved in ZFC:
$$
\cont^+ = \A(\ext)\le\A(\ac)=\A(\conn)=\A(\da)\le \A(\pc)\le 2^\cont,
$$
$$
\cont^+ \le \A(\sz)\le 2^\cont.
$$
For more details see \cite{milcie}, \cite{cie-nat},
\cite{cie-rec}, and \cite{natkaniec}.

The main result of the paper is the following theorem.

\thm{main}{

\begin{description}

\item[(1)] {\rm (MA)} $ \add(\da,\sz) \ge \add(\ac,\sz) \ge \omega $. 

\item[(2)] {\rm (MA)} $\add(\sz,\ac)= \add(\sz,\da) = \cont$.

\item[(3)] If the theory ``ZFC + $\exists$ measurable cardinal'' is consistent then so is ``ZFC + $ \add(\ac,\sz) > \cont > \omega_1 $.'' 

\item[(4)] $\add(\pc,\sz)=\A(\sz)$ and $\add(\sz,\pc)= 2^{\cont}$.

\end{description}
}


The following remains an open problem. (See Fact~\ref{either-ch}.)

\pr{pr1}{Does the equality $\add(\ac,\sz) = \omega$ hold in 
``ZFC + MA'' (or in ``ZFC + CH''?)}

Let us make here some comments about the theorem. Parts (1)
and (3) give only lower bound for $\add(\ac,\sz)$. So one may wonder whether
it is possible to give in ZFC any non-trivial upper bound for that number.
However, in the model used to prove (3) it is possible to have $\cont^+=2^\cont$,
so it cannot be proved in ZFC that $\add(\ac,\sz) < 2^{\cont}$. But it is
unknown whether $\add(\ac,\sz)\le \cont^+$ in ZFC. The next comment is about symmetry of $\add$. It is consistent that $\A(\sz)<2^\cont$. (See~\cite{cie-nat}.) Hence the part (4) implies that $\add$ is not symmetric in general.

Next we give some corollaries of the main result. To state the first one, note that $-\sz=\{-f \colon f \in \sz \} = \sz $. This observation, Proposition~\ref{prop1} and the part (2) of Theorem~\ref{main} immediately imply the following corollary.

\cor{sum_acsz}{{\rm (MA)} Every function $f\colon \real \to \real$ can be 
represented as a sum of almost continuous and Sierpi{\'n}ski-Zygmund
functions.}

Let us mention that the corollary, so also the parts (1) and (2)
of Theorem~\ref{main}, 
cannot be proved in ZFC alone (i.e., without any additional assumptions.)
Indeed, if $\real^\real=\ac+\sz$ then there exists an almost continuous
function which is also Sierpi{\'n}ski-Zygmund. An example of a model
with no Darboux (so also almost continuous) Sierpi{\'n}ski-Zygmund function is given in \cite{balcienat}. Hence we can state 

\cor{consistent}{The equalities $\real^\real=\ac + \sz$ and $\real^\real=\da + \sz$ are independent of ZFC.}
 
One may ask whether Corollary~\ref{sum_acsz} can be improved 
by replacing the family $\ac$ of almost continuous functions 
by the family $\ext$ of extendable functions. However, it cannot be done.
The reason is that every extendable function is continuous on some 
perfect set. (See \cite{cie-jas}.) The above observation implies  

\fact{sz-ext}{$ \add(\ext,\sz)=\add(\sz,\ext)=1$.}

One may also try to generalize Corollary~\ref{sum_acsz} 
for all functions from $\real^n$ into $\real$.
However, in the case $n \ge 2$ it can be proved in ZFC that there is 
no almost continuous function which is also Sierpi{\'n}ski-Zygmund.
We have the following remark.

\rem{ac*sz=empty}{Let $n \ge 2$. Then $\ac(\real^n) \cap \sz(\real^n)=\emptyset$
and $$ \add(\ac(\real^n),\sz(\real^n)) = \add(\sz(\real^n),\ac(\real^n)) = 1.$$  }

\proof For every $n \ge 2$ if $f \in \ac(\real^n)\cap \sz(\real^n)$ then
$f|\real^2 \in \ac(\real^2)\cap \sz(\real^2)$. (See \cite{natkaniec}.) 
Hence it is enough to prove the remark for $n=2$. We construct the family 
$\{ B_y: y \in \real \}$ of $\cont$-many blocking sets in $\real^3$ 
with pairwise disjoint $xy$-projections and whose union is the graph of a 
continuous function. Let $B_y=\{ \la x,y,\tan(x) \ra \colon 
x \in ( \frac{-\pi}{2}, \frac{\pi}{2})\} $ for $y \in \real$. Every 
almost continuous function from $\real^2$ to $\real$ must intersect all sets 
$B_y$. Thus it cannot be of Sierpi{\'n}ski-Zygmund type, since it agrees
with the function $F(x,y)=\tan(x)$ on a set of cardinality of continuum.

%\noindent
The second part of the conclusion follows from Proposition~\ref{prop1} (4).
\qed

Let us make here a comment about $ \add(\da(\real^n),\sz(\real^n))$.
It is easy to see that
$\sz(\real^n) \cap \da(\real^n)=\emptyset$ because for each non-constant Darboux function $f \colon \real^n \to \real$ there exists a real number $y$ such that $f^{-1}(y)$ disconnects $\real^n$. Based on this we obtain 
\rem{da*sz}{$\add(\da(\real^n),\sz(\real^n)) = \add(\sz(\real^n),\da(\real^n)) = 1$.}  

The next two theorems describe the function $\add$ for other pairs of classes  
considered in this paper.

\thm{other_add1}{Let ${\cal F} \in \{\ext, \ac, \conn, \da, \pc\}$ and 
${\cal F}_1,{\cal F}_2 \in \{\ac, \conn, \da\}$. The following equalities hold.
\begin{description}

\item[(i)] $\add(\C,{\cal F})=\add({\cal F},\C)=1$.

\item[(ii)] $\add({\cal F},\ext)=\A(\ext)=\cont^+$ and $\add(\ext,{\cal F})=\A({\cal F})$.

\item[(iii)] $\add({\cal F},\pc)=\A(\pc)=2^\cont$.

\item[(iv)] $\add({\cal F}_1,{\cal F}_2)=\A(\da)$.

\end{description}
}

\thm{other_add2}{ Let ${\cal F} \in \{\ext, \ac, \conn, \da, \pc, \sz \}$.
The following holds.

\begin{description}

\item[(i)] $\add(\ad,\ac)=\add(\ad,\conn)=\add(\ad,\da)=\A(\ac)$.

\item[(ii)] $\add(\ad,\ext)=\A(\ext)=\cont^+$.

\item[(iii)] $\add(\ad,\pc)=\A(\pc)=2^\cont$.

\item[(iv)] $\add(\ad,\sz) > \cont$.

\item[(v)] $\add({\cal F},\ad)=\A(\ad)=2$ and $\add(\C,\ad)=\add(\ad,\C)=1$.

\end{description}
}

We state here next open problem.
\pr{pr2}{Does $\add(\ad,\sz)$ equal to $\A(\sz)?$}


The paper is organized as follows. The proof of Theorem~\ref{main} is presented in next three sections.
The proof of parts (1)-(2) is given in Section~\ref{sec2}. It is based on 
two auxiliary results (Lemmas~\ref{theof} and ~\ref{finite}) which are of interest on their own.
The proofs of parts (3) and (4) are presented in Sections~\ref{sec3}
and~\ref{sec4}, respectively. In Section~\ref{sec5} we prove 
Theorems~\ref{other_add1} and~\ref{other_add2}. 

\section{Proof of Theorem~\ref{main} (1)-(2)} \label{sec2} 

 We begin this section with presenting two lemmas. To state the lemmas we
need the following definitions. For $X \sq \real$ by $\ccont(X)$ we denote
the family of all functions $f \colon X \to \real $ which can be represented
as a union of less than $\cont$-many partial continuous functions.
To simplify notation we write $\ccont$ and $\Ccont$
for $\ccont(\real)$ and $\bigcup_{X \sq \real} \ccont(X)$, respectively.
Observe that under the assumption of regularity of $\cont$ (so also under MA)
$\sz(X) + \ccont(X)=\sz(X) $ and $\sz(Y) \cap \ccont(Y)= \emptyset$
for any $X, Y \sq \real$ with $|Y|=\cont$. The same assumption about $\cont$
implies also that the union of any family $F \sq \Ccont$ of cardinality less
than $\cont$ contains a function from $\ccont(\bigcup_{f \in F}\dom(f))$.

Now we introduce the next definition. Let $A \sq \real$ be everywhere of second 
category, that is $A \cap I $ is of second category for every nontrivial 
interval $I$. We define ${\cal F}_A$ as a family of all 
$F\subseteq \real^{\real}$ whose union $\bigcup F$ contains no function from
$\ccont(A \cap B)$ for any non-meager Borel set $B$. That is
$$
{\cal F}_A=\left \{ F \subseteq \real^{\real} \colon \forall 
B \in ({\cal B} \setminus {\cal M}) \;\; \forall f \in \ccont(A \cap B)  
\;\; f \nsubseteq \bigcup F \right \}.
$$
\lem{theof}{{\rm (MA)} Let $F \in {\cal F}_A$ be a family such that 
$|F| < \A(\sz)$. There exists a $g \in \sz(A) $ such that every extension
$\bar{g} \colon \real \to \real$ of $g$ is almost continuous and
$g+F \subseteq \sz(A) $.} 

\proof 
Let $\la f_{\alpha}: \alpha < \cont \ra $ be a sequence of all continuous 
functions defined on $G_{\delta}$ subsets of $\real$. 

(1) First we construct a partial real function $g^{\prime} \in \SZ$ 
with $\dom(g^{\prime}) \sq A$ such that 
for every $f \in F, \; g^{\prime}+f \in \SZ$ and any extension of
$g^{\prime}$ on $\real$ is in $\ac$.
We do this by transfinite induction. We construct a sequence
$\langle g_{\xi} : \xi < \cont \rangle$ of partial real functions satisfying 
the following conditions for every $\alpha  < \cont$:

\begin{description}
\item[(a)]
$ D_{\alpha} =\dom(g_\alpha)$ is countable;

\item[(b)]
$g_{\alpha} \mbox{ is dense subset of } (f_{\alpha}|A) \setminus 
\bigcup_{\xi < \alpha} \left ( f_{\xi} \cup (D_{\xi}\times \real) \cup \bigcup (f_{\xi}-F) \right )$.
\end{description}

Notice that $D_{\alpha} \cap D_{\beta}=\emptyset$ and $D_{\alpha} \sq A$ 
for $\alpha < \beta < \cont$.
Now we define $g^{\prime}=\bigcup_{\xi < \cont}g_{\xi}$. We will show that
$g^{\prime}$ has the required properties. 

\begin{description}
\item[(i)]
$g^{\prime}, g^{\prime}+f \in \SZ $, for every $f \in F$.

Let $\xi < \cont$. We see from the condition (b) that 
$[g^{\prime}=f_{\xi}], \; [(g^{\prime}+f)=f_{\xi}]
\sq \bigcup_{\alpha \le \xi} D_{\alpha}$.
Hence $|[g^{\prime}=f_{\xi}]|, |[(g^{\prime}+f)=f_{\xi}]| \le \xi \omega < \cont$.

\item[(ii)]
Any extension of $g^{\prime}$ is an almost continuous function.

We will prove that $g^{\prime}$ intersects every blocking set $B \sq \real$. 
$B$ contains a continuous function $q$ defined on a Borel set of second category.
(See~\cite{kellum}.) Let $\alpha_B$ be the smallest ordinal number such that $f_{\alpha_B}$ agrees 
with $q$ on a set residual in some interval $J \sq \dom(B)$. $B$ is closed and therefore 
$f_{\alpha_B} | J \sq B$.
From the definition of $\alpha_B$ and MA we see that 
$\bigcup_{\xi < \alpha_B} [f_{\xi}=q]$ is of first category as the union of less than $\cont$-many sets of first category.
Recall that $F \in {\cal F}_A$. This implies that 
$ (I \cap A) \setminus \bigcup_{\xi < \alpha_B} \bigcup_{f \in F}[(f_{\xi}-f)=q] $ 
is of second category for every nontrivial interval $I$. 
The above holds because otherwise we would have that
$ (K \cap A) \sq \bigcup_{\xi < \alpha_B} \bigcup_{f \in F}[(f_{\xi}-f)=q] $
for some $K \in {\cal B} \setminus {\cal M}$. Then for every $x\in (K \cap A)$
there are $\xi< \alpha_B$ and $f \in F$ such that $f_{\xi}(x)-f(x)=q(x)$.
Define $h\colon (K \cap A) \to \real$ by $h(x)=f_{\xi}(x)-q(x)=f(x)$.
It is easy to see that $h$ is a subset of both 
$\bigcup_{\xi < \alpha_B} (f_{\xi}-q)$ and $\bigcup F$. In particular, it
implies that $h\in \ccont(K \cap A)$ which contradicts the assumption that
$F \in {\cal F}_A$.

Hence $ (J \cap A) \setminus 
\bigcup_{\xi < \alpha_B} (\bigcup_{f \in F}[(f_{\xi}-f)=q] \cup [f_{\xi}=q] \cup D_{\xi})$ 
is of second category. Therefore $D_{\alpha_B} \cap J \ne \e$. 
This implies $g^{\prime} \cap B \supseteq g_{\alpha_B} \cap B \ne \e$ 
($g_{\alpha_B}$ and $f_{\alpha_B}$ coincide on $D_{\alpha_B} \cap J$).
\end{description}

(2) Let $g{''}:A \setminus \dom(g^{\prime}) \rightarrow \real$ be a
Sierpi{\'n}ski-Zygmund function such that $g{''}+F \subseteq  \SZ$. 
Such a function exists because $|F| < \A(\sz)$. We define $g=g^{\prime} \cup g{''}$. 
We see that $g \in \sz(A)$, any extension of $g$ onto $\real$ is in $\ac$, 
and $g+F \subseteq \sz(A)$.
\qed

\lem{finite}{{\rm (MA)} Let $\{ f_i \}_1^n \subseteq \real^{\real}$, $n=1,2, \dots $.
There exists $\{ f_i^{\prime} \}_1^n \in {\cal F}_A$ such that 
$f_i | A_i \in \ccont(A_i)$, where $A_i=[ f_i \ne f_i^{\prime} ]$. }

\proof
The proof is by induction on number $n$ of functions.

Assume that the lemma is true for every 
$\{ g_i \}_1^{n-1} \subseteq \real^{\real}, \; n \ge 1$. Let us fix
$\{ f_i \}_1^n \subseteq \real^{\real}$. We will construct a family
$\{ f_i^{\prime} \}_1^n \in {\cal F}_A$ such that 
$f_i | [ f_i \ne f_i^{\prime} ] \in \ccont([ f_i \ne f_i^{\prime} ])$ for all
$i\le n$.

We start with showing that the following claim holds for 
all $f, h, h^{\prime} \in \real^\real$.
$$
\mbox{If }
f|[ f \ne h] \in \Ccont \mbox{ and } h|[ h \ne h^{\prime}] \in
\Ccont \mbox{ then } f|[ f \ne h^{\prime}] \in \Ccont.
$$
This is so because we have that $[f \ne h^{\prime}] \sq 
[f \ne h] \cup [h \ne h^{\prime}]$
and consequently 
$$
f|[ f \ne h^{\prime}]\sq
f|([f \ne h] \cup [h \ne h^{\prime}])=
f|[f \ne h] \; \cup \; f|([h \ne h^{\prime}]
\setminus [f \ne h]) \sq 
$$ 
$$ \sq f|[f \ne h] \; \cup \; 
h|[ h \ne h^{\prime}].
$$
This completes the proof of the claim.

Now observe that, by the inductive assumption, there exists
$\{ h_i\}_2^n \in {\cal F}_A$ such that
$f_i|[f_i\not= h_i] \in \Ccont$ for $i=2, \dots , n$. Put $h_1=f_1$.
If $\{ h_i^\prime \}_1^n \in {\cal F}_A$ is such that 
$h_i|[h_i\not= h_i^\prime] \in \Ccont$ for $i=1, \dots , n$ then,
based on the above claim, also $f_i|[f_i\not= h_i^\prime] \in \Ccont$ for all
$i$. So without loss of generality we may assume that 
$\{ f_i \}_2^n \in {\cal F}_A$.

Next we define the family ${\cal B}_{f_1,\dots ,f_n}$ as follows
$${\cal B}_{f_1,\dots ,f_n}=\{ A \cap B \colon B \in {\cal B} \setminus {\cal M}
\; \&  \; \exists {f \in \ccont(A \cap B)} \; f \subseteq \bigcup f_i \}.$$ 
There exists a maximal element $A_{\max}$ in 
${\cal B}_{f_1,\dots ,f_n}$ with respect to the relation $\subseteq^{\ast}$ 
defined by
$$X_1 \subseteq^{\ast} X_2, \mbox{ if } X_1 \setminus X_2 \mbox{ is of
first category. }$$ 
To prove the existence let us consider
${\cal S}=\{B \in {\cal B} \setminus {\cal M} \colon 
A \cap B \in {\cal B}_{f_1,\dots ,f_n}\}$.
For every $B \in {\cal S}$ we define a maximal open set $U_B$ such that $B$ is 
residual in $U_B$. Since $\real$ has a countable base, there is a sequence 
$\la B_n \in {\cal S} \colon n < \omega \ra$ such that 
$\bigcup_{B \in {\cal S}} U_B = \bigcup_{n < \omega} U_{B_n}$.
We claim that $A_{\max}=\bigcup_{n < \omega} (A \cap B_n)$ is the desired 
maximal element. First we notice that $A_{\max} \in 
{\cal B}_{f_1,\dots ,f_n}$. Now, let $A \cap B \in {\cal B}_{f_1,\dots ,f_n}$. 
From the properties of the sets $B_n \; (n < \omega)$ we get that
$B \sq^\ast U_B \sq \bigcup_{n < \omega} U_{B_n} \subseteq^{\ast}
\bigcup_{n < \omega} B_n$. So $A \cap B \subseteq^{\ast} A_{\max}$.

Now, let $f$ be the function associated with $A_{\max}$ (e.g. $f \in 
\ccont(A_{\max})$ and $f \subseteq \bigcup f_i$).
The function $f$ can be represented as $f=\bigcup f_i|A_i$, where 
$\bigcup_{i \le n} A_i=A_{\max}$, $A_i \cap A_j=\emptyset \; (i \ne j)$, and 
$f_i|A_i \in \ccont(A_i)$.
Let us consider the following functions $f_i^{\prime}=f_i|{(\real \setminus A_i)} \cup g_i$,
where $g_i \in \sz(A_i)$ $(i=1, \dots, n)$. 
We will show that $\{ f_i^{\prime}\}_1^n$ is the required family, that is
$\{ f_i^{\prime}\}_1^n \in {\cal F}_A$.
Assume, by contradiction, that $\{ f_i^{\prime}\}_1^n \notin {\cal F}_A$.
Thus there exists a set $A^{\prime}$ of the form 
$A \cap B$ for some $B \in {\cal B} \setminus {\cal M}$ such that
$A^{\prime}=\bigcup A_i^{\prime}$, $A_i^{\prime}$ are pairwise disjoint and 
$f_i^{\prime}|A_i^{\prime} \in \ccont(A_i^{\prime})$. Let us denote  
$\bigcup (f_i^{\prime}|A_i^{\prime})$ by $f^{\prime}$.
Note that $A^{\prime} \sq^{\ast} A_{\max}$.
Since $g_1 \in \sz(A_1)$, we have
$|A_1 \cap A_1^{\prime}|< \cont $. This observation and Martin's Axiom imply that $A_1 \cap A_1^{\prime} \in {\cal M}$. 
So we may assume $A_1 \cap A_1^{\prime}= \emptyset$. 
Then $f^{\prime}|(A_1 \cap A^{\prime}) \sq \bigcup_{i=2}^n f_i$.
This implies that $f^{\prime}|(A_1 \cap A^{\prime}) \;
\cup  \; f|(\bigcup_{i=2}^n A_i \cap A^{\prime}) \in \ccont(A^{\prime})$.
Hence $\bigcup_{i=2}^n f_i$ contains a function from $\ccont(A^{\prime})$.
So $\{ f_i\}_2^n \not\in {\cal F}_A$. Contradiction.
\qed


Before we show how the above two lemmas imply parts (1) and (2) of the main 
result, let us make a remark regarding Lemma~\ref{finite}. One could expect
the lemma to hold for bigger families of functions. However, 
Lemma~\ref{finite} cannot be generalized for infinite families of 
functions. Let us see the following counterexample.

\ex{not-inf}{{\rm (CH)} There exists an infinite family 
$\{ f_n \}_{n < \omega} \subseteq \real^{\real}$ for which the conclusion of 
Lemma~\ref{finite} fails.}

\proof Continuum Hypothesis implies the existence of an Ulam matrix on $\real$, e.g. the family 
$\{M^n_{\xi} : n < \omega , \xi < \cont\}$ of subsets of $\real$ with
$$M^n_{\xi} \cap M^n_{\alpha} = \emptyset \mbox{, for } n<\omega, \; \xi < \alpha < \cont,$$
$$\mbox{the complement of } \bigcup_{n<\omega} M^n_{\xi} \mbox{ is a countable set, for } \xi <  \cont.$$
Fix an enumeration $\{ x_{\xi}: \xi < \cont \}$ of $\real$. Define
$f_n$ as an extension of $\bigcup_{\xi < \cont} x_{\xi}\chi_{M^n_{\xi}}$ onto 
$\real$, for every $n < \omega$. We are now in a 
position to show that $F=\{f_n \colon n < \omega\}$ is the counterexample for 
the conclusion of Lemma~\ref{finite}.
Since every vertical section of $\bigcup F$ is countable and every horizontal section is comeager,
it follows that $\bigcup F$ is non-Borel set of second category. Now, let $A_n \sq \real$
be such that $f_n | A_n \in \ccf(A_n)$, for every $n$. Since 
the graph of a continuous function is meager in $\real^2$, we obtain that 
$\bigcup_{n < \omega} f_n | A_n$
is also meager as a union of countably many meager sets. 
We conclude from this that there exists a meager horizontal section of 
$\bigcup_{n < \omega} f_n | A_n$. Therefore the set
$\bigcup F \setminus \bigcup_{n < \omega} f_n | A_n$ contains a constant function defined
on comeager Borel set. 
\qed

Using very similar technique as the above we can prove 

\fact{either-ch}{ {\rm (CH)} Either $\add(\ac,\sz)=\omega$ or 
$\add(\ac,\sz) > \cont$.}

\proof Let us assume that $F=\{\phi_\xi \colon \xi < \cont \} \sq \real^\real$ 
witnesses $\add(\ac,\sz) \le \cont$. For every $n < \omega$, define a function 
$f_n^\ast$ as an extention of $\bigcup_{\xi < \cont} \phi_{\xi}\chi_{M^n_{\xi}}$ onto $\real$, 
where $\{M^n_{\xi} : n < \omega , \xi < \cont\}$ is an Ulam matrix.
We claim that $\{ f_n^\ast \colon n < \omega \}$ witnesses
$\add(\ac,\sz) \le \omega$. To see this fix an $h \in \ac$. By our assumption 
about $F$, there exists an $\xi_0 < \cont $ such that $h+f_{\xi_0}\not\in \sz$. That means 
$h+f_{\xi_0}$ is continuous on a set $X$ of cardinality continuum. Since 
$\real \setminus \bigcup_{n<\omega} M^n_{\xi_0}$ is countable we obtain that 
$|X \cap M^m_{\xi_0}|=\cont$ for some $m < \omega$. Hence $h+f_m^\ast$ 
is continuous on a set of cardinality continuum which means that
$h+f_m^\ast \not\in \sz$. 
\qed

\noindent
{\bf Proof of } $ \add(\ac,\sz) \ge \omega $ (under MA).

We begin by fixing $F=\{f_1, \dots, f_n \} \sq \real^{\real}$. 
Let $F^{\prime}=\{f_1^{\prime}, \dots, f_n^{\prime} \} \in {\cal F}_{\real}$ be 
a corresponding family given by Lemma~\ref{finite} for $A=\real$. 
Based on Lemma~\ref{theof}, we can find a
$g \in \ac \cap \sz$ such that $g+F^{\prime} \sq \sz$.
Since $f_i|[ f_i^{\prime} \ne f_i ] \in \Ccont$ and   
$g \in \sz$, we obtain that $g+f_i \in \sz$ (for $i=1, 2, \dots$, n.)
\qed

In order to prove part (2) of Theorem~\ref{main} we need to state one more
lemma.

\lem{add(sz,dar)}{$\add(\sz,\da) \le 2^{< \cont }$.}

\proof
Let us consider the following family of functions 
${\cal F}^{< \cont} = \{ r\chi_A \colon A \in [\real]^{< \cont }, r \in 
\rational \}$.
Obviously $|{\cal F}^{< \cont}|=2^{< \cont }$.
We claim that $$\forall_{g \in SZ} \; g+{\cal F}^{< \cont} \not \subset 
\da.$$
To see this, fix $g \in \sz$. Let $r_0 \in \rational $ such that 
$\inf g < r_0 < \sup g$. Then $g-r_0 \chi_A \not\in \da$, 
where $A=g^{-1}[r_0]$.
\qed

\noindent
{\bf Proof of }$\add(\sz,\ac)= \add(\sz,\da) = \cont$ (under MA).

Since $\add(\sz,\ac) \le \add(\sz,\da)$ and 
$\add(\sz,\da) \le 2^{< \cont }=\cont$ (assuming MA), it is sufficient 
to prove that for every family $F \sq \real^{\real}$ of cardinality less than $\cont$ there exists
a Sierpi{\'n}ski-Zygmund function $h\colon \real \rightarrow \real$ 
satisfying the property $h+F \sq \ac$.

Let $F=\{f_{\xi} \colon \xi < \kappa\} \sq \real^{\real}$ $(\kappa = |F| < \cont)$ and 
$\{A_{\xi} \colon \xi < \kappa\}$ be a partition of $\real$ into Bernstein sets.
By Lemma~\ref{finite}, for every $\xi < \kappa $ we can find a function
$f_{\xi}^{\prime}$ such that  the singleton $\{f_{\xi}^{\prime}\}$ belongs to
${\cal F}_{A_{\xi}}$ and
$f_{\xi}^{\prime}|[ f_{\xi}^{\prime} \ne f_\xi ] \in \Ccont$. Now, applying Lemma
\ref{theof} for every ${\xi} < \kappa$ we obtain a sequence 
$\la g_{\xi} \colon A_{\xi} \rightarrow \real : \xi < \kappa \ra$ for which the 
following holds
$$g_{\xi}+ f_{\xi}^{\prime} \in \SZ 
\mbox{ and any extension of $g_{\xi}$ on $\real$ is in $\ac$, for } \xi < \kappa.$$
Since $f_{\xi}^{\prime}|[ f_{\xi}^{\prime} \ne f_{\xi} ] \in \Ccont$ and 
$\sz(X)+\ccont(X)=\sz(X)$ for every $X \sq \real$, we conclude that
$g_{\xi}+ f_{\xi} \in \SZ$, $\xi < \kappa$.
Put $h=\bigcup_{\xi < \kappa} -(g_{\xi}+ f_{\xi})$. Since Martin's Axiom implies the regularity of $\cont$ we obtain that $h \in \sz$.
Clearly, $h+F \sq \ac$.
\qed

As the final remark let us notice that parts (1) and (2) of 
the main result as well as Lemmas~\ref{theof} and~\ref{finite} could be proved 
under weaker assumptions. The proofs require only two consequences of Martin's 
Axiom: $\cont=\cont^{< \cont}$ (this implies regularity of $\cont$); the union of 
less than $\cont$-many meager sets is meager.


\section{Proof of Theorem~\ref{main} (3)} \label{sec3} 

We will show that the existence of 
$\cont$-additive $\sigma$-saturated ideal ${\cal J}$ in $P(\real)$ containing ${\cal M}$
implies $\add(\ac,\sz) > \cont$. It is known that the existence of such an ideal 
is equiconsistent with ``ZFC $+ \; \exists $ measurable cardinal.''\footnote{
The desired model is obtained by adding $\kappa$-many Cohen reals, where $\kappa$ is a 
measurable cardinal in the ground model.} 
(See \cite{kam}.)

First notice that we may assume that ${\cal J} \cap {\cal B}={\cal M}$. 
To see this suppose that there exists a Borel set $B$ of second category in 
${\cal J} $. $B$ is residual in some open interval $I$. Then 
$I \in {\cal J}$ because $I \setminus B$ is meager and 
$I=(B \cap I) \cup (I\setminus B)$. Now, let $U$ be a maximal open set belonging
to ${\cal J}$. Such a set exists because the union of all open sets
from ${\cal J}$ can be represented as a union of countable many such sets.
We have that $\real \setminus U$ contains a nonempty open interval $I_0$. 
Otherwise it would be nowhere-dense and then 
$\real=U \cup (\real \setminus U) \in {\cal J}$. Now, any homeomorphism between
$I_0$ and $\real$ induces the desired ideal on $\real$.

The schema of the proof is similar to the idea of combining Lemmas~\ref{theof} and
\ref{finite} in the proof of $\add(\ac,\sz) \ge \omega$. First step is to show 
that 

\begin{description}
\item[$(\ast)$]
for each $f\colon \real \to \real$ there exists an $f^{{\cal J}} \in 
\real^\real$ such that $f|[f \not= f^{{\cal J}}] \in \CC$ and 
$f^{{\cal J}}|X \notin \ccf(X)$ for every $X \notin {\cal J}$. 
\end{description}

To see this fix an $f \in \real^\real$. 
We claim that there exists a set $Y$ 
such that $f|Y \in \ccf(Y)$ and 
$Y^{\prime} \sq^{{\cal J}} Y$ for all $Y^{\prime}$ satisfying $f|Y^{\prime} \in \ccf(Y^{\prime})$, where $\sq^{{\cal J}}$ is defined by
$$Z_1 \subseteq^{{\cal J}} Z_2, \mbox{ if } Z_1 \setminus Z_2 \in {\cal J}.$$

If the claim did not hold then we could easily construct a strictly 
increasing (in terms of $\sq^{{\cal J}}$) uncountable sequence of subsets of $\real$. 
Indeed, assume that the desired sequence of sets $X_{\xi}$ is defined for all $\xi < \alpha$, where $\alpha < \omega_1$. Note that $f|\bigcup_{\xi < \alpha} X_{\xi} \in \CC$. By assumption there exists a set $X$ such that $ \bigcup_{\xi < \alpha} X_{\xi} \sq^{{\cal J}} X \not\sq^{{\cal J}} \bigcup_{\xi < \alpha} X_{\xi}$ and $f|X \in \CC$. We set $X_{\alpha}=X$. Thus by transfinite induction the  sequence is defined
for all $\alpha < \omega_1$.
But the existence of this sequence would imply the existence of an uncountable family of disjoint
sets outside of ${\cal J}$ which contradicts the fact that ${\cal J}$ is $\sigma$-saturated. 

So we proved that the set $Y$ exists. Now put $f^{{\cal J}}=f|(\real\setminus Y) \cup g$,
where $g$ is any function from $\sz(Y)$. Clearly,
$f^{{\cal J}}$ is the desired function from $(\ast)$.

 In the next step we fix a family $F$ of real functions of cardinality $\cont$.
Let $F=\{h_{\xi} \colon \xi < \cont\}$ be an enumeration of $F$ and 
 $\la f_{\alpha}: \alpha < \cont \ra $ be a sequence of all continuous 
functions defined on $G_{\delta}$ subsets of $\real$. 
Based on the previous reasoning we may assume that 
$h_{\xi}|X \notin \ccf(X)$ for every $X \notin {\cal J}$ and $\xi < \cont$.
Notice that if $\gamma, \alpha < \cont$ and $f_{\alpha}|X \sq 
\bigcup_{\xi,\beta < \gamma} (f_{\xi}-h_{\beta})$ then $X\in {\cal J}$. 
This is so since $X \sq \bigcup_{\xi,\beta < \gamma} 
[f_{\alpha}=f_{\xi}-h_{\beta}]$ and every set 
$[f_{\alpha}=f_{\xi}-h_{\beta}] = [h_{\beta}=f_{\xi}-f_{\alpha}] \in {\cal J}$. Consequently, the set $\dom(f_{\alpha} \setminus 
\bigcup_{\xi,\gamma < \alpha}(f_{\xi}-h_{\gamma}) )$ does not belong to ${\cal J}$ provided $\dom(f_{\alpha}) \not\in {\cal J}$.

Now we construct a sequence 
$\langle g_{\xi} : \xi < \cont \rangle$ of partial functions such that 
$$g_{\alpha} \mbox{ is a countable dense subset of } f_{\alpha} \setminus 
\bigcup_{\xi,\gamma < \alpha}((f_{\xi}-h_{\gamma}) \cup f_{\xi} \cup L(D_{\xi}))
\mbox{ for } \alpha  < \cont,$$ 
\noindent
where $D_{\gamma}=\dom(g_{\gamma})$.

The same kind of argument as in the proof of Lemma~\ref{theof} (i)$\&$(ii) shows that 
$g^{\prime}=\bigcup_{\xi<\cont} g_{\xi}$ is in $\SZ$ and
intersects every blocking set. So if $g$ is any Sierpi{\'n}ski-Zygmund 
extension of $g^{\prime}$ then $g \in \ac$ and $g+F \sq \sz$.
\qed

\section{Proof of Theorem~\ref{main} (4)} \label{sec4}

First we prove $\add(\pc,\sz)=\A(\sz)$. In order to do it we need the 
following straightforward lemma.

\lem{pc-dense}{For every function $f \in \real^{\real} $ there is a function $f^{\prime} \in \pc$
such that $|[f\not = f^{\prime}]|\le \omega$.}

\proof Let $g \colon \rational \to \rational$ be a function with dense graph. Then 
$f^{\prime}=g\cup f|({\real \setminus \rational})$ is the required function.
\qed

Now, to show $\add(\pc,\sz)=\A(\sz)$, let us notice that 
$\add(\pc,\sz) \le \add(\real^\real,\sz)=\A(\sz)$. What is left to prove is
that $\add(\pc,\sz) \ge \A(\sz)$. Let $F \sq \real^\real$ be a family of 
cardinality less than $\A(\sz)$. So there exists a function $g \in \real^\real$ 
such that $g+F \sq \sz$. Let $g^{\prime}\in \pc$ be a function obtained from 
$g$ by applying Lemma~\ref{pc-dense}. Since every Sierpi{\'n}ski-Zygmund 
function modified on a set of cardinality less than $\cont$ remains 
Sierpi{\'n}ski-Zygmund, it is easy to see that $g^{\prime}+F \sq \sz$. 



Before we start proving that $\add(\sz,\pc)= 2^{\cont}$, we introduce the 
following 
\defi{defsz-set}{A set $X \subseteq \real^2$ is called 
{\it Sierpi{\'n}ski-Zygmund\/} set 
(shortly {\it SZ-set\/}),
if for every partial real continuous function $f$ we have
$|f \cap X| < {\cont}$.}
An argument, similar to the one used in proving the existence of 
Sierpi{\'n}ski-Zygmund function, leads to 

\lem{sz-set}{There exists an SZ-set $X \subseteq \real^2$ such that 
$|\real \setminus X_x|<{\cont}$ for every $x \in \real$, where 
$X_x=\{y \in \real \colon \la x,y \ra \in X\}$.}

\proof Let $\la x_{\alpha}: \alpha < \cont \ra $ and
$\la f_{\alpha}: \alpha < \cont \ra $ 
be the sequences of all real numbers and all continuous functions 
defined on a $G_{\delta}$ subset of $\real$, respectively. We will
define the set $X$ by defining its vertical sections by transfinite induction.
For every $\alpha < \cont$ we put
$$X_{x_{\alpha}}=\real\setminus \{f_{\xi}(x_{\alpha}) \colon \xi < \alpha \}.$$ 
Put $X=\bigcup_{\alpha < \cont } \{x_{\alpha} \} \times X_{x_{\alpha}}$. 
It is obvious that $X$ has the required properties.
\qed

\cor{densesz}{There exists a family 
$\{Q_x \subseteq \real \colon x \in \real\}$ of pairwise disjoint countable 
dense sets such that $\bigcup \prod_{x \in \real}Q_x$ is an SZ-set. }
The next lemma is proved in ~\cite{cie-rec}.
							   
\lem{cie-rec}{\mbox{\rm \cite[Lemma 2.2]{cie-rec}} If $B\subseteq \real$ has 
cardinality ${\cont}$ and $H\subseteq \rational^B$ is such that 
$|H|<2^{\cont}$
then there is a $g\in \rational^B$ such that $h\cap g\ne\emptyset$
for every $h\in H$.}
We give more general version of this lemma.
\lem{c-r_gen}{If $B\subseteq \real$ has cardinality ${\cont}$ and
$H\subseteq \prod_{x \in B}Q_x$ is such that $|H|<2^{\cont}$
then there is a $g\in \prod_{x \in B}Q_x$ such that $h\cap g\ne\emptyset$
for every $h\in H$.}
\proof For every $x \in B$ let $f_x \colon Q_x \to \rational$
be a bijection. Now, for each $h \in H$ we define $h^{\prime}$ as follows
$$h^{\prime}(x)=f_x(h(x)) \mbox{ for all } x \in B.$$ 
The family $H^{\prime}=\{h^{\prime} \colon h \in H\} \sq \rational^B$ 
has cardinality less than $2^{\cont}$. Thus, by Lemma~\ref{cie-rec}, 
there is a function $g^{\prime} \in \rational^B$ intersecting every 
element of $H^{\prime}$. Put $g(x)=f_x^{-1}(g^{\prime}(x))$, 
for all $x \in B$. It is clear that $g \in \prod_{x \in B}Q_x$ and 
$h\cap g\ne\emptyset$ for every $h\in H$.
\qed 

\noindent
{\bf Proof of $\add(\sz,\pc)=2^\cont$.} 

The proof follows the idea of the proof of \cite[Theorem 1.7 (3)]{cie-rec}.
Let $F\sq\real^\real$ be such that $|F|<2^{{\cont}}$.
We will find a $g \in \sz$ such that $g+ F \sq \pc$. 

Let $\G$ be the family of all triples $\la I,p,m\ra$ where
$I$ is a nonempty open interval with rational end-points,
$p\in\rational$, and $m<\omega$.
For each $\la I,p,m\ra\in\G$ 
define a set $B_{\la I,p,m\ra}\sq I$ of size ${\cont}$ such that 
$B_{\la I,p,m\ra}\cap B_{\la J,q,n\ra}=\e$ for any distinct
$\la I,p,m\ra$ and $\la J,q,n\ra$ from $\G$. 

Let $\la I,p,m\ra\in\G$ be fixed. For each $f\in F$ 
choose $h^f_{\la I,p,m\ra}\in \prod_{x \in B_{\la I,p,m\ra}} Q_x$
such that
\[
\left |p-\left (f(x)+h^f_{\la I,p,m\ra}(x)\right ) \right |< \frac{1}{m}\ 
\mbox{ for every } x\in B_{\la I,p,m\ra}.
\]
Then, by Lemma~\ref{cie-rec} used with a set
$H_{\la I,p,m\ra}=\left \{h^f_{\la I,p,m\ra}\colon f\in F \right \}$,
there exists a
$g_{\la I,p,m\ra}\in \prod_{x \in  B_{\la I,p,m\ra}} Q_x$
such that 
\[
\forall f\in F\ \exists x\in B_{\la I,p,m\ra}\ 
h^f_{\la I,p,m\ra}(x)=g_{\la I,p,m\ra}(x).
\]
Now, let $g \in \prod_{x \in \real} Q_x$ be a common extension
of all functions $g_{\la I,p,m\ra}$. 
Corollary~\ref{densesz} implies that $g$
is of Sierpi{\'n}ski-Zygmund type. The function $g$ has also the following 
property. For every $\la I,p,m\ra\in\G$ and every $f\in F$ there exists 
$x\in  B_{\la I,p,m\ra}\sq I$ such that
\[
\left |p-\left (f(x)+g(x)\right ) \right | < \frac{1}{m}.
\]
So, each function $f+g$, for $f \in F$, is dense in $\real^2$.
Thus $f+g\in \pc$. \qed


\section{Proofs of Theorems~\ref{other_add1} and~\ref{other_add2}} \label{sec5}

In this section we present proofs of 
Theorems~\ref{other_add1} and~\ref{other_add2}.
Before we do this, let us recall some definitions and cite some theorems. 
Let $h \in \ext$. We say that a set $G\subset\real$  
is {\em $h$-negligible \/}
provided $f \in\ext$ for every 
function $f\colon \real \to \real$ for which $f=h$ on a set 
$\real\setminus G$.
For a cardinal number $\kappa\leq\cont$, 
a function $f\colon \real \to \real $ is called $\kappa$ {\it strongly Darboux\/} 
if $f^{-1}(y)$ is $\kappa$-dense. If $\kappa=\omega$ then we simply say 
that $f$ is strongly Darboux.
We denote the family of all $\kappa$ strongly Darboux functions by $\da(\kappa)$. 
It is obvious from the definition that
$$ \da(\lambda)\sq
\da(\kappa)\ \mbox{ for all cardinals }\ 
\kappa\leq\lambda\leq\cont.$$

We also introduce the family $\da(\per)$ of {\em perfectly Darboux }functions
as the class of all functions $f \colon \real \to \real$ such that 
$Q \cap f^{-1}(y) \not= \emptyset$ for every perfect set $Q \sq \real$ and $y \in \real$. In other words, a function $f$ is 
perfectly Darboux if for every $y \in \real$ $f^{-1}(y)$ is a Bernstein set. 
Notice that $\da(\per) \sq \da(\kappa)$ for every $\kappa \leq \cont$.

The following theorem is proved in~\cite{milcie}.
\thm{milcieth1}{$\A(\ac)=\A(\da)=\A(\da(\omega_1))$.}

A little modification of the proof of the above theorem gives the following lemma.

\lem{milciegen}{Let ${\cal F} \in \{\ad, \ext\}$. Then $\add({\cal F},\ac)= \add({\cal F},\da)$. }

The proof of Lemma~\ref{milciegen} requires the use of the following lemma and proposition.  

\lem{gendar}{Let $X$ be any set of cardinality continuum and $F \sq \real^X$ satisfies the condition $|F| < \A(\da)$. There exists a $g \colon X \to \real$ such that $(g+f)^{-1}(y)\not= \emptyset$ for each $y \in \real$.}
\proof Let $b \colon \real \to X$ be a bijection. By Theorem~\ref{milcieth1} and monotonicity of $\A$ we have that $\A(\da)=\A(\da(\omega))$. Hence we can find a 
$g^\prime \colon \real \to \real $ satisfying the property that
$g^\prime+ (f \circ b) \in \da(\omega)$ for each $f \in F$. Put $g=g^\prime \circ b^{-1}$. Clearly, g is the desired function.
\qed

\prop{prop2}{$\A(\da)=\A(\da(\per))$.}
\proof Fix a family $F \sq \real^\real$ of cardinality less than $\A(\da)$. Next,let $\{B_\xi \colon \xi < \cont \}$ and $\{ P_\xi \colon \xi < \cont \}$
be a family of pairwise disjoint Bernstein sets and an enumeration of all
perfect subsets of $\real$, respectively. We define the sequence
$\la A_\xi \colon \xi < \cont \ra$ by $A_\xi=B_\xi \cap P_\xi$. Obviously
the sets $A_\xi$ are pairwise disjoint and each one of them has cardinality $\cont$. Applying Lemma \ref{gendar} for every $\xi < \cont$ separately, we get a sequence of functions $\la g_\xi \colon A_\xi \to \real \;
| \; \xi < \cont \ra$ such that for every $\xi < \cont$ the following holds 
$$\forall f \in F \;\; \forall y \in \real \;\;\; (g_\xi + f)^{-1}(y)\not= \emptyset.$$ Now, if $g \in \real^\real$ is any extension of
$\bigcup_{\xi < \cont} g_\xi$ onto $\real$ then $g+F \sq \da(\per)$.
\qed 


\noindent
{\bf Proof of Lemma~\ref{milciegen}.}

 First we show that
\begin{description}
\item[$(\ast \ast)$]
$\add({\cal F},{\cal F}_0) > \cont$ for 
${\cal F}_0 \in \{\ac, \da(\omega_1)\}$.
\end{description}
Let us fix a family $F \sq \real^\real$ with cardinality $\cont$. To prove the case ${\cal F}=\ad$
consider a $\cont$-dense Hamel basis $H$. There exists a partition $\{B_f \colon f\in F  \}$ of $H$ into $\cont$-dense sets. Since the projection of every 
blocking set in $\real^2$ contains an interval, we can find, for every $f \in F$, 
a partial function $g_f \colon B_f \to \real$ such that $g_f + f$ intersects every 
blocking set in at least $\omega_1$ points. Thus every extension of 
$g_f + f$ onto $\real$ is almost continuous and $\omega_1$ strongly Darboux. If $g \in \real^\real$ is any function
containing $\bigcup_{f \in F} g_f$ then $g+F \sq \ac \cap \da(\omega_1)$. 
In particular, we can choose
$g$ to be an additive function. Hence $\add(\ad,{\cal F}_0) > \cont$ for 
${\cal F}_0 \in \{\ac, \da(\omega_1)\}$.

Now consider the case ${\cal F}=\ext$. If ${\cal F}_0=\ac$ then we have 
the inequality $\add(\ext,\ac)\ge \add(\ext,\ext)=
\A(\ext) = \cont^+ > \cont$ which follows from Proposition~\ref{prop1} (2)$\&$(5). Now, let us focus on the case ${\cal F}_0=\da(\omega_1)$.
Let $Q \sq \real $ be $\cont$-dense meager $F_{\sigma}$-set.
Then, according to~\cite[Proposition 4.3]{cie-jas},
there exists an extendable function $f\colon\real\to\real$  such that 
the set $\real \setminus Q$ is $f$-negligible.
Since $|F| < \A(\da)=\A(\da(\per))$, there exists a function $h \in \real^\real$ such that $h + F \sq \da(\per)$. Notice here that any perfectly Darboux function modified on a meager set is in $\da(\omega_1)$. This implies that the function $g=f|Q \; \cup \; h|(\real \setminus Q)$ shifts $F$ into $\da(\omega_1) \sq \da$. Since $Q \sq [f=g]$ we have that $g \in \ext$. Observe also that $F$ could be any family with
$|F| < \A(\da)=\A(\da(\per))$. So we actually proved that 
$$\add(\ext,\da) \ge \add(\ext,\da(\omega_1))\ge \A(\da).$$ 
This finishes the proof of $(\ast \ast)$.

Now the argument follows the schema of the proof of Theorem~\ref{milcieth1}.\footnote{
For reader's convenience, we include this slight modification of the proof
from~\cite{milcie} in this paper.} We start with proving the equality
$\add({\cal F},\da)= \add({\cal F},\da(\omega_1))$. Obviously $\add({\cal F},\da) \ge 
\add({\cal F},\da(\omega_1))$.
To justify the other inequality
let $\kappa=\add({\cal F},\da(\omega_1))$. By $(\ast \ast)$ we get that $\kappa > \cont$.
We will show that $\kappa\ge \add({\cal F},\da)$.

Consider a family $G\sq\real^\real$ of cardinality $\kappa$
witnessing $\kappa=\add({\cal F},\da(\omega_1))$.
We define a new family $G^*=\{h\in\real^\real\colon \;\exists g\in G\; h=^*g\}$,
where $h=^*f$ if and only if $|\{x\colon h(x)\neq f(x)\}| \le \omega$.
Notice here that $|G^*|=\kappa$. This is so because 
$\kappa>\cont$ and for every $f\in\real^\real$ the set 
$\{h\in\real^\real\colon h=^*f\}$ has cardinality $\cont$. We claim that 
$G^*$ witnesses $\kappa \ge \add({\cal F},\da)$. Indeed, let $f \in {\cal F}$. Then, by the choice of $G$, there exists a $g \in G$ satisfying the following $f+g \notin \da(\omega_1)$.
This implies the existence of a non-trivial closed interval $I$ and $y \in \real$ for which 
$|I  \cap (f+g)^{-1}(y)| \le \omega$. By modification of $g$ on a countable set, we get a function $g^* \in G^*$
with the property that $(f+g^*)[I] \cap (-\infty,y) \not= \emptyset \not= (f+g^*)[I] \cap (y,\infty)$ and $y \notin (f+g^*)[I]$. Therefore $(f+g^*) \notin \da$. This ends the proof of the equality $\add({\cal F},\da)= \add({\cal F},\da(\omega_1))$. 
 
What remains to show is that $\add({\cal F},\ac)= \add({\cal F},\da(\omega_1))$.
The inequality $\add({\cal F},\ac) \le \add({\cal F},\da)=\add({\cal F},\da(\omega_1))$ is obvious, so we just need to prove 
that $\add({\cal F},\ac) \ge \add({\cal F},\da(\omega_1))$. This time consider $K\sq\real^\real$ witnessing $\add({\cal F},\ac)=\lambda$. We put 
$K^*=\{g-h_B \colon g\in K \mbox{ and $B$ is a blocking set}\}$, where $h_B \in \real^\real$
is a function such that $h_B|\dom(B) \sq B$. Clearly $|K^*|=\lambda$ because there are only continuum many blocking sets and $\lambda > \cont$.  
Let $f\in {\cal F}$. Then, by the choice of $K$,
there exist a $g\in K$ and a blocking set $B$ such that
$(f+g)\cap B=\emptyset$. In particular,
\[
[f+(g-h_B)]\cap(B-h_B)=[(f+g)\cap B]-h_B=\emptyset,
\]
where
we define $Z-h_B=\{(x,y-h_B(x))\colon (x,y)\in Z\}$
for any $Z\sq\real^2$.
From the definition of $h_B$ we have $\dom(B)\times\{0\} \sq (B-h_B)$. 
Thus $[f+(g-h_B)]\cap[\dom(B)\times\{0\}]=\emptyset$. This means that 
$f+(g-h_B) \not\in\da(\omega_1)$, since $\dom(B)$ contains a non-trivial interval.
But $g-h_B\in K^*$, so $K^*$ witnesses  $\lambda \ge \add({\cal F},\da(\omega_1))$.
This finishes the proof of $\add({\cal F},\ac)= \add({\cal F},\da(\omega_1))$ as well as
whole Lemma~\ref{milciegen}.
\qed

\noindent
{\bf Proof of Theorem~\ref{other_add1}.}

(i) Notice that it is enough to show (i) for ${\cal F}=\pc$ since $\add(\C,{\cal F})\le \add(\C,\pc)$ by Proposition~\ref{prop1} (1). To see that  
$\add(\C,\pc)=\add(\pc,\C)=1$ observe that $\C+\pc=\pc$. Therefore, if $f\not\in \pc$ then there is no $g\in \C$ such that $g+f \in \pc$.


(ii)  The first part follows from the inequality $$\A(\ext) \ge \add({\cal F},\ext)\ge \add(\ext,\ext)=\A(\ext)=\cont^+,$$ 
where the first equality is implied by Proposition~\ref{prop1} (5).  

To see $\add(\ext,{\cal F})=\A({\cal F})=\A(\ac)$ for ${\cal F} \in \{\ac, \conn, \da\}$ let us note that, by Lemma~\ref{milciegen} and Proposition~\ref{prop1} (2), 
$\add(\ext,\ac)=\add(\ext,\conn)=\add(\ext,\da)$. Finally, the desired equality follows from
$\add(\ext,\da) \ge \A(\da)$, which is shown in the prove of $(\ast \ast)$ in Lemma~\ref{milciegen}.  

The proof of the case $\add(\ext,\pc)=\A(\pc)=2^\cont$ will be given in (iii). 
 
(iii) Again, by the monotonicity of $\add$, it suffices to show (iii) for ${\cal F}= \ext$.  
Let $Q \sq \real $ and $f\colon \real \to \real$ be as in the proof of $(\ast\ast)$ Lemma~\ref{milciegen}, i.e., $Q$ is $\cont$-dense meager $F_\sigma$-set and $f$ is an extendable function such that $\real \setminus Q$ is $f$-negligible.
Fix a family $F \sq \real^\real$ of cardinality less than $2^\cont$.
Now, a small modification in the proof of the equality $\add(\sz,\pc)=2^\cont$ in
Section~\ref{sec4} (the sets $B_{\la I,p,m\ra}$ can be chosen to be subsets of $\real \setminus Q$), 
gives us a function $g \colon \real \to \real$ which shifts $F$ into $\pc$ and which
agrees with $f$ on the set containing $Q$. In particular, $g$ is
an extendable function.

(iv) The last part of Theorem~\ref{other_add1} is proved by the following inequality
$$\A(\da)=\A(\ac)= \add(\ac,\ac) \le \add({\cal F}_1,{\cal F}_2) \le \add(\da,\da) = \A(\da).$$ 
\qed

\noindent
{\bf Proof of Theorem~\ref{other_add2}.}

(i) To prove the first part of Theorem~\ref{other_add2} we need one more lemma.

\lem{lowerb2}{$\add(\ad,\da) \ge \A(\da(\per))$. In particular, $\add(\ad,\da)=\A(\da)$.}

\proof Let $P \sq \real$ be a perfect set with the property that 
$P \cup \{1\}$ is linearly independent over $\rational$. Observe 
that for every 
$p,q \in \rational,$ $p \not\in \{0,1\}$ we have $(pP+q) \cap P=\emptyset$. 
Now, consider a countable partition 
$\{P_n \colon n < \omega \}$ of $P$ into perfect sets.
Using this partition and the above observation we can easily construct a family $\{P_n^{\star} \colon n < \omega \}$ of disjoint perfect sets such that $\bigcup_{n < \omega}P_n^\star$ is independent
over $\rational$ and for every nontrivial interval $I \sq \real$ there
is an $m < \omega$ such that $P_m^\star \sq I$. 
Note that $\bigcup_{n < \omega}P_n^\star$ is a $\cont$-dense meager 
$F_\sigma$-set.
 
To prove the inequality $\add(\ad,\da) \ge \A(\da(\per))$ let us fix
a family $F\sq \real^\real$ such that $|F|<\A(\da(\per))$. There exists
a function $g \in \real^\real$ satisfying the property 
$g+F \sq \da(\per)$. We claim that if $g^{\star}\colon \real \to \real$ is any additive extension
of $g|\bigcup_{n < \omega}P_n^\star$ then $g^{\star}+F \sq \da$. More precisely, for every $f \in F$, $g^{\star} + f$ is strongly Darboux. To see this pick any $f\in F$, $y \in \real$, and any interval 
$I$. There exists $m < \omega$  such that $P_m^\star$ is contained in $I$. Furthermore, we can find $x \in P_m^\star \sq I$ for which
$g^{\star}(x)+f(x)=g(x)+f(x)=y$. This shows that $g^{\star} + f$ is strongly Darboux.

The second statement in the lemma is proved by the obvious inequality
$\A(\da) \ge \add(\ad,\da) \ge \A(\da(\per))$ and Proposition~\ref{prop2}.
\qed

Now, (i) follows from Lemmas~\ref{milciegen},~\ref{lowerb2}, and Proposition~\ref{prop1} (1).

(ii) Since $\add(\ad,\ext) \le \A(\ext)=\cont^+$, it suffices to show the inequality $\add(\ad,\ext) \ge \cont^+$. So for every $F=\{f_\xi \colon \xi < \cont\} \sq \real^\real$ we need to find 
a $g \in \ad$ such that $g+F \sq \ext$.
 
Let $\la D_\xi \colon \xi < \cont \ra$ be a sequence of pairwise disjoint 
$\cont-$dense meager $F_\sigma$ sets such that $\bigcup_{\xi < \cont} D_\xi$ is linearly
independent over $\rational$. Such a sequence can be constructed in a similar way as the $\cont-$dense meager $F_\sigma$-set 
in the proof of Lemma \ref{lowerb2}. Now, by {\rm \cite[Proposition 4.3]{cie-jas}}, 
for every $\xi < \cont$ we can find $h_\xi \in \ext$ such that $\real\setminus D_\xi$ is $h_\xi$-negligible.
We define $g$ as an additive extension of $\bigcup_{\xi < \cont} (h_\xi-f_\xi)|D_\xi$. 

To see that $g+f_\xi \in \ext$ for every $\xi$, observe that $g+f_\xi=h_\xi$ on $D_\xi$. But the set $\real\setminus D_\xi$ is $h_\xi$-negligible. So each $g+f_\xi$ is extendable.


(iii) The prove of this part is similar to the prove of Theorem~\ref{main} (4).
Fix a Hamel basis $H$ which is a Bernstein set. By choosing the sets   
$B_{\la I,p,m\ra}$ to be subsets of $H$, we can obtain, 
for a given family $F$ of real functions with cardinality less than $2^\cont$,
an additive function which shifts $F$ into $\pc$.

(iv) Let us fix a family $F=\{h_\xi \colon \xi < \cont\} \sq \real^\real$ and a Hamel basis $H=\{x_\xi \colon \xi < \cont\}$. We will construct an additive function $g$ with the property that $g+F \sq \sz$, by defining it on $H$ using induction. For a given $\alpha < \cont$, we choose
$$g(x_\alpha) \notin \left ( \bigcup_{q \in \rational} \bigcup_{\xi, \gamma < \alpha} q(f_\gamma - h_\xi)[\lin(x_\beta \colon \beta \le \alpha)] \right )+ g[\lin(x_\beta \colon \beta < \alpha)],$$
where $\la f_{\alpha}: \alpha < \cont \ra $ is a sequence of all continuous functions defined on $G_{\delta}$ subsets of $\real$. Such a choice is possible because the cardinality of the considered set is less than $\cont$. This choice also assures that $g+F \sq \sz$. To see that observe the following  
$[g+h_\xi=f_\alpha]=[g=f_\alpha -h_\xi] \sq \lin(x_\beta \colon \beta < \alpha)$ for all $\alpha, \xi < \cont$. Thus $|[g+h_\xi=f_\alpha]| = \omega \alpha < \cont$, which proves that $g+h_\xi \in \sz$.
 
(v) First observe that $\A(\ad)=2$. This follows from Proposition~\ref{prop1} (3)\&(5)
and obvious equality $\ad -\ad =\ad$. Recall also that $\add({\cal F},\ad) \le \A(\ad)$
and ${\cal F} - \ad=\ad - {\cal F}={\cal F}+ \ad$ for all ${\cal F} \in    
\{\ext, \ac, \conn, \da, \pc, \sz \}$. Thus, by Proposition~\ref{prop1} (3)
and Theorem~\ref{other_add2} (i)-(iv), we get that ${\cal F}+ \ad=\real^\real$.
Consequently, $\add({\cal F},\ad)=2$.

The same part of Proposition~\ref{prop1} implies the second statement in (v). 
This is so because $\C -\ad=\ad - \C \not= \real^\real$. 
The characteristic function of a point, say $\charf{\{0\}}$, is an example 
of a function witnessing the above property. Indeed, $(\charf{\{0\}} + \C) 
\cap \ad = \emptyset$ because every additive function is either continuous
or has a dense graph (see {\rm \cite[Exercise 4, Section 7.3]{cie}}.)
\qed

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\end{document}

