%Typeset with Latex format %correction 02/23/2001 \documentclass{article} \usepackage{amsmath,amssymb} \newcommand{\forces}{\mathrel{\|}\joinrel\mathrel{-}} \newcommand{\suc}{{\rm succ}} \newcommand{\cf}{{\rm cf}} \newcommand{\cc}{{\rm con}} \newcommand{\ccc}{{\rm con}} \newcommand{\supp}{{\rm supp}} \newcommand{\nor}{{\rm norm}} \newcommand{\norsup}{{\rm\underline{norm}}} \newcommand{\sq}{\subseteq} \newcommand{\real}{{\mathbb R}} %\newcommand{\real}{{\Re}} \newcommand{\R}{{\real}} \newcommand{\rational}{{\mathbb Q}} \newcommand{\Q}{{\rational}} \newcommand{\integer}{{\mathbb Z}} \newcommand{\N}{{\mathbb N}} \newcommand{\Z}{{\mathbb Z}} \newcommand{\NN}{{\cal N}} \newcommand{\nnn}{{\rm N}} \newcommand{\Sg}{{\Sigma}} \newcommand{\s}{{\sigma}} \newcommand{\h}{{\aleph}} \newcommand{\la}{{\langle}} \newcommand{\ra}{{\rangle}} \newcommand{\restr}{{\hbox{$\,|\grave{}\,$}}} \newcommand{\srestr}{{\hbox{${\scriptstyle\,|\grave{}\,}$}}} \newcommand{\lin}{{\rm Lin_{\rational}}} \newcommand{\A}{{\rm A}} \newcommand{\add}{{\rm Add}} \newcommand{\M}{{\cal M}} \newcommand{\F}{{\cal F}} \newcommand{\G}{{\cal G}} \newcommand{\C}{{\rm C}} \newcommand{\B}{{\cal B}} \newcommand{\T}{{\cal T}} \newcommand{\D}{{\cal D}} \newcommand{\E}{{\cal E}} \newcommand{\e}{{\emptyset}} \newcommand{\ep}{{\varepsilon}} \newcommand{\g}{{\gamma}} \renewcommand{\d}{{\delta}} \renewcommand{\H}{{\cal H}} \renewcommand{\P}{{\cal P}} \newcommand{\ext}{{\rm Ext}} \newcommand{\conn}{{\rm Conn}} \newcommand{\Ccont}{{\rm C_{\rm part}^{< \cont}}} \newcommand{\ccont}{{\rm C^{< \cont}}} \newcommand{\ad}{{\rm AD}} \newcommand{\ac}{{\rm AC}} \newcommand{\da}{{\rm D}} \newcommand{\pc}{{\rm PC}} \newcommand{\ccf}{{\rm CC}} \newcommand{\sd}{{\rm SD}} \newcommand{\sz}{{\rm SZ}} \newcommand{\CC}{{\rm CC_{part}}} \newcommand{\SZ}{{\rm SZ_{part}}} \newcommand{\const}{{\rm Const}} \newcommand{\bor}{\mbox{${\cal B}or$}} \newcommand{\bd}{{\rm bd}} \newcommand{\per}{{\rm P}} % characteristic function \newcommand{\charf}[1]{\mbox{\raise.48ex\hbox{$\chi$}$_{#1}$}} %continuum %\newcommand{\cont}{{\bf c}} \newcommand{\cont}{{\mathfrak c}} \def\dom{{\rm dom}} \def\proof{\noindent {\sc Proof. }} \def\qed{\hfill\vrule height6pt width6pt depth1pt\medskip}%{\hfill$\Box$} %% Theorems, etc. \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{problem}[theorem]{Problem} \newtheorem{example}[theorem]{Example} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{Fact}[theorem]{Fact} \newcommand{\thm}[2]{\begin{theorem}$\!\!\!${\bf .} \label{#1}{\sl #2}\end{theorem}} \newcommand{\cor}[2]{\begin{corollary}\label{#1}{\sl #2}\end{corollary}} \newcommand{\prop}[2]{\begin{proposition}\label{#1}{\sl #2}\end{proposition}} \newcommand{\lem}[2]{\begin{lemma}\label{#1}{\sl #2}\end{lemma}} \newcommand{\pr}[2]{\begin{problem}\label{#1}{\rm #2}\end{problem}} \newcommand{\ex}[2]{\begin{example}\label{#1}{\sl #2}\end{example}} \newcommand{\defi}[2]{\begin{definition}\label{#1}{\rm #2}\end{definition}} \newcommand{\rem}[2]{\begin{remark}\label{#1}{\rm #2}\end{remark}} \newcommand{\fact}[2]{\begin{Fact}\label{#1}{\rm #2}\end{Fact}} \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 \begin{thebibliography}{99} \bibitem{balcienat} M. Balcerzak, K. Ciesielski, T. Natkaniec, {\emph {Sierpi{\'n}ski-Zygmund functions that are Darboux, almost continuous, or have a perfect road}}, Arch. Math. 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