%\documentstyle[11pt,amstex]{amsarth} \documentstyle[11pt,amstex]{amsart} %\input{polish} \textheight 19.4cm \textwidth 12.6cm \voffset -1.7cm \hoffset 2.4cm \setcounter{page}{143} \setcounter{section}{1} %\def \L{\char147} %\def \l{\char149} %\usepackage{amstex}\usepackage{amssymb} %\def\preceq{\leq} %\def\prec{<} \date{}%Version of June 9, 1997} %\input{epsf.tex} \newsymbol \varsubsetneq 2320 \newcommand{\forces}%{\Vdash} {\mathrel {{\vrule height 6.9pt depth -0.1pt}\! \vdash }} \newcommand{\restriction}%{\rest}{{\restriction}} {{\hbox{$\,|\grave{}\,$}}} \renewcommand{\dot}{\overset{\boldsymbol .}} % Standard Spaces %\def\integer{\mathbb Z} %\def\natural{\mathbb N} %\def\rational{\mathbb Q} %\def\real{\mathbb R} \def\natural{{\Bbb N}} \def\rational{\Bbb Q} \def\real{{\Bbb R}} \def\reals{{\real}} \def\R{{\real}} %\def\complex{\mathbb C} \def\complex{{\Bbb C}} \def\complexnumb{\complex} \def\su{\subseteq} \def\mathcal{\cal} %End of proof marker %\def\qed{\hfill$\Box$} %Proof %\def\proof{\noindent{\sc Proof.} } %\def\pf{\proof} \newcommand{\supp}{\operatorname{supp}} \newcommand{\cl}{\operatorname{cl}} \newcommand{\bd}{\operatorname{bd}} \newcommand{\id}{\operatorname{id}} %% Theorems, etc. \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{problem}{Problem} \newcommand{\thm}[2]{\begin{theorem}\label{#1}#2\end{theorem}} \newcommand{\cor}[2]{\begin{corollary}\label{#1}#2\end{corollary}} \newcommand{\examp}[2]{\begin{example}\label{#1}#2\end{example}} \newcommand{\lem}[2]{\begin{lemma}\label{#1}#2\end{lemma}} \newcommand{\prop}[2]{\begin{proposition}\label{#1}#2\end{proposition}} \newcommand{\pr}[2]{\begin{problem}\label{#1}#2\end{problem}} %%%%%%%%%%%%%%%%%%%%% MISCELLANEOUS \renewcommand{\P}{{\cal P}} \newcommand{\A}{{\cal A}} \newcommand{\B}{{\cal B}} \newcommand{\C}{{\cal C}} \newcommand{\D}{{\cal D}} \newcommand{\G}{{\cal G}} \newcommand{\I}{{\cal I}} \newcommand{\U}{{\cal U}} \newcommand{\V}{{\cal V}} \newcommand{\T}{{\cal T}} \newcommand{\e}{{\varepsilon}} \renewcommand{\d}{{\delta}} \newcommand{\M}{{\cal M}} \newcommand{\N}{{\cal N}} \newcommand{\calN}{\N} \newcommand{\Lin}{{\cal L}} \newcommand{\F}{{\cal F}} \newcommand{\calF}{\F} \newcommand{\add}{{\op{A}}} \newcommand{\genadd}{{\op{GA}}} \newcommand{\mul}{{\op{M}}} \newcommand{\diff}{{\frak e}} \newcommand{\same}{{d}} \newcommand{\CIVP}{{\op{CIVP}}} \newcommand{\SCIVP}{{\op{SCIVP}}} \newcommand{\WCIVP}{{\op{WCIVP}}} \newcommand{\ordinarytop}{\T_{\cal O}} % characteristic function \newcommand{\charf}[1]{\mbox{\raise.48ex\hbox{$\chi$}$_{#1}$}} \def\qed{\hfill$\Box$ \medskip} %\def\qed{\hfill\vrule height4pt width4pt depth1pt\medskip} \def\conn{{\operatorname{Conn}}} \def\ext{{\operatorname{Ext}}} \def\pc{{\operatorname{PC}}} \def\ac{{\operatorname{AC}}} \def\cf{\op{cf}} \def\continuum{{\frak c}} \def\co{\continuum} \def\cuum{\continuum} \def\la{\langle} \def\ra{\rangle} \newcommand{\non}{\operatorname{non}} \newcommand{\cov}{\operatorname{cov}} \newcommand{\cout}{{\operatorname{C}_{out}}} \newcommand{\cin}{{\operatorname{C}_{in}}} \newcommand{\Const}{{\op{Const}}} \newcommand{\dec}{\op{dec}} \newcommand{\AT}{{@@}} \newcommand{\op}[1]{\operatorname{#1}} \title [\it Set theoretic real analysis ] {\vspace {1.0cm}\uppercase {\Large \bf {Set Theoretic Real Analysis}}} \author[K. Ciesielski]{\uppercase {\bf K. Ciesielski$^*$}} %\address{} \keywords{Fubini theorem, continuous images, continuous restrictions, Continuum Hypothesis, Martin's Axiom, Blumberg Theorem, symmetric continuity, Peano curve, Darboux functions, cardinal functions, addition of functions, derivatives, extensions of measure} \subjclass{Primary: 26--02, 04--02; secondary 54--02, 28--02} \thanks{$^*$The author wishes to thank Professors Lee Larson, Marek Balcerzak, Andy Bruckner, Tomasz Natkaniec, Juris Stepr\={a}ns, and Brian Thomson for reading preliminary versions of this paper and helping in improving its final version. \newline This work was partially supported by NSF Cooperative Research Grant INT-9600548 with its Polish part financed by KBN.} \thanks{\ \\ ISSN 1425--6908 \quad \copyright Heldermann Verlag} \dedicatory{\footnotesize {\it Received February 25, 1997 and, in revised form, September 8, 1997}} %\date{February 25, 1997} \begin{document} \leftline{\footnotesize \sc Journal of Applied Analysis} \leftline{\footnotesize Vol. 3, No. 2 (1997), pp. 143--190} \vspace{3cm} \centerline{\it Special invited paper} \maketitle \thispagestyle{empty} \bigskip \begin{abstract} This article is a survey of the recent results that concern real functions (from $\real^n$ into $\real$) and whose solutions or statements involve the use of set theory. The choice of the topics follows the author's personal interest in the subject, and there are probably some important results in this area that did not make it to this survey. Most of the results presented here are left without proofs. \end{abstract} \bigskip \section*{\bf 1. Historical background} \medskip The study of real functions has played a fundamental role in the development of mathematics over the last three centuries. The seventeenth century discovery of calculus by Newton and Leibniz was largely due to increased understanding of the behavior of real functions. %The discovery of calculus by %eighteenth century mathematicians, notably Newton and Leibniz, was largely due %to increased understanding of the behavior of real functions. The birth of analysis is often traced to the early nineteenth century work of Cauchy, who gave precise definitions of concepts such as continuity and limits for real functions. Convergence problems while approximating real functions by Fourier series gave rise to both the Riemann and Lebesgue integrals. Cantor developed his set theory in an effort to answer uniqueness questions about Fourier series \cite{KL,BKR,Th}. During this time, different techniques have been used as the theory behind them became available. For example, after Cauchy, various limiting operations such as pointwise and uniform convergence were studied, giving rise to various approximation techniques. At the turn of this century, measure theoretic techniques were exploited, leading to stochastic convergence ideas in the 1920's. Also, at about the same time topology was developed, and its applications to analysis gave rise to functional analysis. In recent years, a new research trend has appeared which indicates the emergence of a yet another branch of inquiry that could be called {\em set theoretic real analysis}. This area is the study of families of real functions using modern techniques of set theory. These techniques include advanced forcing methods, special axioms of set theory such as Martin's axiom~(MA) and proper forcing axiom~(PFA), as well as some of their weaker consequences like additivity of measure and category. (See \cite{Ku}, \cite{Sh}, \cite{Fr}, \cite{DalWood}, and \cite{BJ} for examples of this work.) Set theoretic real analysis is closely allied with descriptive set theory, but the objects studied in the two areas are different. The objects studied in descriptive set theory are various classes of (mostly nice) {\bf sets} and their hierarchies, such as Borel sets or analytic sets. Set theoretic real analysis uses the tools of modern set theory to study {\bf real functions} and is interested mainly in more pathological objects. Thus, the results concerning subsets of the real line (like the series of studies on ``small'' subsets of $\real$ \cite{AMillerSubsetsOfR}, or deep studies of the duality between measure and category \cite{Ox,AMiller:MeasureCat,BJ}) are considered only remotely related to the subject. (However, some of these duality studies spread to real analysis too. For example, see a monograph~\cite{CLO:book}.) Set theoretic real analysis already has a long history. Its roots can be traced back to the 1920's, where powerful new techniques based on the Axiom of Choice~(AC) and the Continuum Hypothesis~(CH) can be seen in many papers from such journals as Fundamenta Mathematicae and Studia Mathematica. The most interesting consequences of the Continuum Hypothesis discovered in this period have been collected in~1934 monograph of Sierpi\'nski, {\it Hypoth\`ese du Continu}~\cite{Si:CH}. The influence of Sierpi\'nski's results (and the monograph) on the set theoretic real analysis can be best seen in the next section. The new emergence of the field was sparked by the discovery of powerful new techniques in set theory and can be compared to the parallel development of set theoretic topology during the late 1950's and 1960's. In fact, it is a bit surprising that the development of set theoretic analysis is so much behind that of set theoretic topology, since at the beginning of the century the applicability of set theory in analysis was at least as intense as in topology. This, however, can be probably attributed to the simple fact, that in the past half of a century there were many mathematicians that knew well both topology and set theory, and very few that knew well simultaneously analysis and set theory. Our terminology is standard and follows ~\cite{Ci}. \medskip \stepcounter{section} \section*{\bf 2. New developments in classical problems} \medskip The first problem we wish to mention here is connected with the Fubini--Tonelli Theorem. The theorem says, in particular, that if a function \linebreak $f\colon[0,1]^2 \to[0,1]$ is measurable then the iterated integrals $\int_0^1\int_0^1 f(x,y)\,dy\,dx$ and $\int_0^1\int_0^1 f(x,y)\,dx\,dy$ exist and are both equal to the double integral \linebreak $\int\int f\, dm_2$, where $m_2$ stands for the Lebesgue measure on $\real^2$. But what happens when $f$ is non-measurable? Then clearly the double integral does not exist. However, the iterated integrals might still exist. Must they be equal? The next theorem, which is a classical example of an application of the Continuum Hypothesis in real analysis, gives a negative answer to this question. \medskip \begin{theorem}[Sierpi\'nski 1920, \cite{Si1}] %2.1 \label{th:sierp1} If the Continuum Hypothesis holds then there exists %a function $f\colon[0,1]^2\to[0,1]$ for which the iterated integrals $\int_0^1\int_0^1 f(x,y)\,dy\,dx$ and $\int_0^1\int_0^1 f(x,y)\,dx\,dy$ exist but are not equal. \end{theorem} \medskip \begin{pf} Let $\preceq$ be a well ordering of $[0,1]$ in order type continuum $\continuum$ and define $A=\{\la x,y\ra\in[0,1]^2\colon x\preceq y\}$. Let $f$ be the characteristic function $\charf{A}$ of $A$. Then for every fixed $y\in[0,1]$ the set $\{x\in[0,1]\colon f(x,y)\neq 0\}= \linebreak \{x\in[0,1]\colon x\preceq y\}$ is an initial segment of a set ordered in type $\continuum$. So, by CH, it is at most countable and \[ \int_0^1\int_0^1 f(x,y)\,dx\,dy=\int_0^1 0\,dy=0. \] Similarly, for each $x\in[0,1]$ the set $\{y\in[0,1]\colon f(x,y)\neq 1\}=\linebreak \{y\in[0,1]\colon y\prec x\}$ is at most countable and \[ \int_0^1\int_0^1 f(x,y)\,dy\,dx=\int_0^1 1\,dy=1. \] Thus, $\int_0^1\int_0^1 f(x,y)\,dy\,dx=1\neq 0=\int_0^1\int_0^1 f(x,y)\,dx\,dy$. \end{pf} \medskip Sierpi\'nski's use of the Continuum Hypothesis in the construction of such a function begs the question whether such a function can be constructed using only the axioms of ZFC. The negative answer was given in the 1980's by Laczkovich~\cite{La}, Friedman~\cite{Fd} and Freiling~\cite{Fl}, who independently proved the following theorem. \medskip \begin{theorem}%2.2 \label{th:sierp1NEG} There exists a model of set theory ZFC in which for every function $f\colon[0,1]^2\to[0,1]$, the existence of the iterated integrals $\int_0^1\int_0^1 f(x,y)\,dy\,dx$ and $\int_0^1\int_0^1 f(x,y)\,dx\,dy$ implies their equality. \end{theorem} \medskip It is also worthwhile to mention that the function $f$ from the proof of Theorem~\ref{th:sierp1NEG} has the desired property as long as every subset of $\real$ of cardinality less than continuum has measure zero, i.e., when the smallest cardinality $\non(\calN)$ of the non-measurable subset of $\real$ is equal to $\continuum$. Since the equation $\non(\calN)=\continuum$ holds in many models of ZFC in which CH fails (for example, it is implied by MA) Theorem~\ref{th:sierp1NEG} is certainly not equivalent to CH. On the other hand, Laczkovich proved Theorem~\ref{th:sierp1NEG} by noticing that: (A) the existence of an example as in the statement of Theorem~\ref{th:sierp1} implies the existence of such an example as in its proof, i.e., in form of $\charf{A}$; (B) there is no set $A\subset[0,1]^2$ with $f=\charf{A}$ satisfying Theorem~\ref{th:sierp1NEG} if $\non(\calN)<\cov(\calN)$, where $\cov(\calN)$ is the smallest cardinality of a covering of $\real$ by the sets of measure zero. (It is well known that the inequality $\non(\calN)<\cov(\calN)$ is consistent with ZFC. See e.g.~\cite{BJ}.) A discussion of a similar problem for the functions $f\colon[0,1]^n\to[0,1]$ and the $n$-times iterated integrals can be found in a~1990 paper of Shipman~\cite{Ship}. The same paper contains also two easy ZFC examples of measurable functions $f\colon [0,1)^2\to\real$ and $g\colon \real^2\to[-1,1]$ for which the iterated integrals exist but are not equal. Thus, the restriction of the above problem to the non-negative functions is essential. \medskip Another classical result arises from a different theorem of Sierpi\'nski of~1928. \medskip \begin{theorem}[Sierpi\'nski \cite{Si2a,Si2b}] %2.3 \label{th:sierp2} If the Continuum Hypothesis \linebreak holds then there exists a set $S\subset\R$ of cardinality continuum such that its image $f[S]\neq [0,1]$ for any continuous $f\colon\R\to[0,1]$. \end{theorem} \medskip The set $S$ from the original proof of Theorem~\ref{th:sierp2} is called {\em Sierpi\'nski set}\/ and it has the property that its intersection $S\cap N$ with any measure zero set $N$ is at most countable.% \footnote{This approach was used in the paper~\cite{Si2a}, while the Luzin set approach in the paper~\cite{Si2b}. Since they are published in the same year, the priority is not completely clear. However in the list of Sierpi\'nski's publications printed in~\cite{Si:Colect} paper~\cite{Si2a} precedes~\cite{Si2b}, suggesting its priority.} Another set that satisfies the conclusion of Theorem~\ref{th:sierp2}, known as {\em Luzin set}\/ (see~\cite{Si2b} or~\cite[property C$_5$]{Si:CH}), is defined as an uncountable subset $L$ of $\real$ whose intersection $L\cap M$ with any meager set $M$ is at most countable.% \footnote{The construction of such a set, under CH, was published by Luzin in~1914~\cite{LusSet}. The same construction had been also published in 1913 by Mahlo~\cite{Mahlo}. But (as is not unusual in mathematics) such a set is commonly known as a Luzin set.} The existence of a Luzin set is also implied by CH. In fact, the constructions of sets $S$ and $L$ under the assumption of CH are almost identical: you list all G$_\delta$ measure zero sets (F$_\sigma$ meager sets) as $\{Z_\xi\colon \xi<\continuum\}$ and define $S$ ($L$, respectively) as a set $\{x_\alpha\colon\alpha<\continuum\}$ where $x_\alpha\in\real\setminus\left(\bigcup_{\xi<\alpha}Z_\xi\right)$. The choice is possible since, by CH, the family $\{Z_\xi\colon\xi<\alpha\}$ is at most countable implying that its union is not equal to $\real$. It is also easy to see that this construction can be carried out if $\cov(\N)=\continuum$ (and its category analog $\cov(\M)=\continuum$ in case of construction of $L$). The sets constructed that way are called generalized Sierpi\'nski and Luzin sets, respectively, and they also satisfy the conclusion of Theorem~\ref{th:sierp2} independently of the size of $\continuum$. Since many models of ZFC satisfy either $\cov(\N)=\continuum$ or $\cov(\M)=\continuum$ (for example, both conditions are implied by MA) it has been a difficult task to find a model of ZFC in which the conclusion of Theorem~\ref{th:sierp2} fails. It has been found by A.~W.~Miller in~1983. \medskip \begin{theorem}[A.~W.~Miller \cite{Mi}]%2.4 \label{th:sierp2NEG} There exists a model of set theory ZFC in which for every subset $S$ of\/ $\real$ of cardinality $\continuum$ there exists a continuous function $f\colon\R\to[0,1]$ such that $f[S]=[0,1]$. \end{theorem} \medskip In his proof of Theorem~\ref{th:sierp2NEG} Miller used the iterated perfect set model, which will be mentioned in this paper in several other occasions. \medskip Some of the most recent set-theoretic results concerning classical problems in real functions are connected with a theorem of Blumberg from~1922. \medskip \begin{theorem}[Blumberg \cite{Bl}] %2.5 \label{th:Blumb} For every $f\colon\R\to\R$ there exists a dense subset $D$ of\/ $\R$ such that the restriction $f\restriction D$ of $f$ to $D$ is continuous. \end{theorem} \medskip The set $D$ constructed by Blumberg is countable. In a quest whether it can be chosen any bigger Sierpi\'nski and Zygmund proved in~1923 the following theorem. \medskip \begin{theorem}[Sierpi\'nski, Zygmund \cite{SZ}]%2.6 \label{th:SZ} There exists a function $f\colon\R\to\R$ whose restriction $f\restriction X$ is discontinuous for any subset $X$ of\/ $\R$ of cardinality $\continuum$. \end{theorem} \medskip Theorem~\ref{th:SZ} immediately implies the following corollary, which shows that there is no hope for proving in ZFC a version of the Blumberg theorem in which the set $D$ is uncountable. \medskip \begin{corollary}[Sierpi\'nski, Zygmund \cite{SZ}] %2.7 \label{cor:SZ} If the Continuum Hy\-po\-the\-sis holds then there exists a function $f\colon\R\to\R$ such that $f\restriction X$ is discontinuous for any uncountable subset $X$ of\/ $\R$. \end{corollary} \medskip The proof of Theorem~\ref{th:SZ} is a straightforward transfinite induction diagonal argument after noticing that every continuous partial function on $\R$ can be extended to a continuous function on a G$_\delta$ set. Corollary~\ref{cor:SZ} raises the natural question about the importance of the assumption of CH in its statement. Is it consistent that the set $D$ in Blumberg Theorem can be uncountable? Can it be of positive outer measure, or non-meager? The cardinality part of these questions is addressed by the following theorem of Baldwin from~1990. \medskip \begin{theorem}[Baldwin \cite{Ba}] %2.8 \label{th:Baldwin} If Martin's Axiom holds then for every function $f\colon\R\to\R$ and every infinite cardinal number $\kappa<\continuum$ there exists a set $D\subset\R$ such that $f\restriction D$ is continuous and $D$ is $\kappa$-dense, i.e., $D\cap I$ has cardinality $\kappa$ for every non-trivial interval $I$. \end{theorem} \medskip Thus under MA the size of the set $D$ is clear. By Theorem~\ref{th:SZ} it cannot be chosen of cardinality continuum (at least for some functions), but it can be always chosen of any cardinality $\kappa$ less than $\continuum$. One might still hope to be able to prove in ZFC that for any $f$ the set $D$ can be found of an arbitrary cardinality $<\continuum$. However, this is false as well, as noticed by several authors: G.~Gruenhage in~1993 (see Rec{\l}aw~\cite[Thm~4]{Recl}) S.~Shelah in~1995 (see~\cite{Sh2}) and the author of this survey (unpublished). \medskip \begin{theorem}[\cite{Recl,Sh2}] %2.9 \label{th:Sh1} There exists a model of ZFC+$\neg$CH (namely a Cohen model) in which there is a function $f\colon\real\to\real$ which is discontinuous on any uncountable subset of\/ $\R$. \end{theorem} \medskip The category version of a question on a size of $D$ has been also settled in the~1995 paper of Shelah~\cite{Sh2} mentioned above. \medskip \begin{theorem}[Shelah \cite{Sh2}] %2.10 \label{th:Sh2} There exists a model of ZFC in which for every function $f\colon\real\to\real$ there exists a set $D\subset\R$ such that $f\restriction D$ is continuous and $D$ is nowhere meager, i.e., $D\cap I$ is non-meager for every non-trivial interval $I$. \end{theorem} \medskip The measure version of the question is less clear. It has been noticed by J.~Brown in~1977 that the precise measure analog of Theorem~\ref{th:Sh2} cannot be proved. (This has been also noticed independently by K.~Ciesielski, whose proof is included below.) \medskip \begin{theorem}[Brown \cite{BrownMeasBl}] %2.11 \label{th:KCBlumb} There exists a function $f\colon\R\to\R$ such that $f\restriction D$ is discontinuous for every set $D\subset\real$ which is nowhere measure zero, i.e., such that $D\cap I$ has positive outer measure for every non-trivial interval $I$. \end{theorem} \medskip \begin{pf} Let $\{F_n\colon n<\omega\}$ be a partition of $\real$ such that $F_0$ is a dense G$_\delta$ set of measure zero and $F_n$ is nowhere dense for each $n>0$. Define $f\colon\real\to\real$ by putting $f(x)=n$ for $x\in F_n$. Now, $f\restriction X$ is discontinuous for any dense $X\subset\real$ which is nowhere measure zero. Indeed, if $X\subset\real$ is dense and nowhere measure zero then there exists an $x\in X\setminus F_0$. Now, if every open set $U$ containing $x$ intersects $F_n\cap X$ for infinitely many $n$ then $f\restriction X$ is discontinuous at $x$. Otherwise, there is an open set $U$ containing $x$ and intersecting only finitely many $F_n$'s. So, we can find a non-empty open interval $I\subset U$ such that $I\cap X\subset F_0$. But this means that $I\cap X$ has measure zero, a contradiction. \end{pf} \medskip However, the following problem asked by Heinrich von Weizs\"acker \cite[Problem AR(a)]{FrProb96} remains open. \medskip \begin{problem} %1 \label{pr:Blum} Is it consistent that every function $f\colon\real\to\real$ is continuous on some set $X\subset\real$ of positive outer measure? \end{problem} \medskip Other generalizations of Blumberg's theorem can be also found in a~1994 survey article~\cite{BrownRestr}. (See also recent papers~\cite{BrownNew} and~\cite{JRblum}.) Another problem that is related in character to the Blumberg's theorem is the following. \begin{quote} {\it Let $\{f_n\colon\real\to[-\infty,\infty]\}_{n=1}^\infty$ be a sequence of arbitrary functions. What is the biggest size of a set $X\subset\real$ for which there exists a subsequence of $\{f_n\}_{n=1}^\infty$ convergent pointwise on $X$?} \end{quote} Clearly such a subsequence can be found for any countable $X\subset\real$. Using this fact Helly~\cite{Hel} proved in~1921 that {\it any bounded sequence of monotone real functions contains a pointwise convergent subsequence}. On the other hand, answering a question of S.~Saks, in~1932 Sierpi\'{n}ski~\cite{Si1932} showed that the Continuum Hypothesis implies the existence of a sequence $\{f_n\colon\real\to[-\infty,\infty]\}_{n=1}^\infty$ such that $\{f_n\restriction X\}_{n=1}^\infty$ has no pointwise convergent subsequence for any uncountable $X\subset\real$. The necessity of additional set theoretical assumptions in the Sierpi\'{n}ski's construction was recently noticed by Fuchino and Plewik~\cite{FP} who showed that the size of $X$ having the property under consideration is characterized by the splitting number ${\bold s}$: {\it For any $X\subset\real$ with $|X|<{\bold s}$ any sequence $\{f_n\colon\real\to[-\infty,\infty]\}_{n=1}^\infty$ has a subsequence convergent pointwise on $X$; however for any $X\subset\real$ with $|X|={\bold s}$ there exists a sequence $\{f_n\colon X\to[0,1]\}_{n=1}^\infty$ with no pointwise convergent subsequence.} (For the definition of the splitting number, see e.g.~\cite{V}.) \medskip In the past few years a lot of activity in real analysis was concentrated around symmetric properties of real functions. (See Thomson~\cite{Th}.) Recall that a function $f\colon\real\to\real$ is symmetrically continuous at $x\in\real$ if \[ \lim_{h\to 0}(f(x+h)-f(x-h))=0, \] and $f$ is approximately symmetrically differentiable at $x$ if there exists a set $S\subset\real$ such that $x$ is a (Lebesgue) density point of $\real\setminus S$ and that the following limit exists \[ \lim_{h\to 0,\, h\notin S}\frac{f(x+h)-f(x-h)}{2h}. \] This limit, which does not depend on the choice of a set $S$, is called {\em the approximate symmetric derivative of $f$ at $x$}\/ and is denoted by $D^s_{ap}f(x)$. We will say that $f$ has a {\em co-countable symmetric derivative at $x$}\/ and denote it by $D^s_c f(x)$ if the set $S$ in the above definition can be chosen to be countable. One of the long standing conjectures (with several incorrect proofs given earlier, some even published) was settled by Freiling and Rinne in~1988 by proving the following theorem. \medskip \begin{theorem}[Freiling, Rinne \cite{FR}] %2.12 \label{th:FR} If $f\colon\R\to\R$ is measurable and such that $D^s_{ap}f(x)=0$ for all $x\in\R$ then $f$ is constant almost everywhere. \end{theorem} \medskip The importance of the measurability assumption in Theorem~\ref{th:FR} was long known from the following theorem of Sierpi\'nski of~1936. \medskip \begin{theorem}[Sierpi\'nski \cite{Si4}] %2.13 \label{th:SiSymm} If the Continuum Hypothesis holds then there exists a non-measurable function $f\colon\R\to\R$ (which is a characteristic function $\charf{A}$ of some set $A$) for which $D^s_c f(x)=0$ for all $x\in\R$. \end{theorem} \medskip In fact, in~\cite{Si4} Theorem~\ref{th:SiSymm} is stated in a bit stronger form% \footnote{Co-countable symmetric derivatives are replaced by co-$<\continuum$ symmetric derivatives and the theorem is proved in ZFC.} from which it follows immediately that the theorem remains true under MA, if the co-countable symmetric derivatives $D^s_c f(x)$ are replaced by the approximate symmetric derivatives $D^s_{ap}(x)$. However, neither Theorem~\ref{th:SiSymm} nor its version with $D^s_{ap}(x)$ can be proved in ZFC. This follows from the following two theorems of Freiling from~1990. \medskip \begin{theorem}[Freiling \cite{Frei:CC}] %2.14 \label{th:Freiling1} If the Continuum Hypothesis fails then for every function $f\colon\R\to\R$ with $D^s_c f(x)=0$ for all $x\in\R$ there exists a countable set $S$ such that $f$ is constant on $\real\setminus S$. \end{theorem} \medskip Thus the existence of a function as in Theorem~\ref{th:SiSymm} is in fact equivalent to the Continuum Hypothesis. \medskip \begin{theorem}[Freiling \cite{Frei:CC}] %2.15 \label{th:Freiling2} It is consistent with ZFC that for every function $f\colon\R\to\R$ with $D^s_{ap} f(x)=0$ for all $x\in\R$ there exists a measure zero set $S$ such that $f$ is constant on $\real\setminus S$. \end{theorem} \medskip More precisely, Freiling proves that the conclusion of Theorem~\ref{th:Freiling2} follows the property that is just a bit stronger than the inequality $\non(\calN)<\cov(\calN)$. (Compare comment following Theorem~\ref{th:sierp1NEG}.) \medskip Another direction in which the symmetric continuity research went was the study of how far symmetric continuity can be destroyed. First note that clearly every continuous function is symmetrically continuous, but not vice versa, since the characteristic function of a singleton is symmetrically continuous. However, it is not difficult to find functions which are nowhere symmetrically continuous. For example, the characteristic function of any dense Hamel basis is such a function.% \footnote{In fact, a Hamel basis $\B$ can be chosen to be both first category and measure zero. Thus $\charf{\B}$ can be measurable and have the Baire property.} How much more can we destroy symmetric continuity? In the non-symmetric case probably the weakest (bilateral) version of continuity that can be defined is the following. A function $f\colon\real\to\real$ is {\em weakly continuous}\label{pageWC} at $x$ if there are sequences $a_n\nearrow0$ and $b_n\searrow0$ such that \[ \lim_{n\to\infty}f(x+a_n)=f(x)=\lim_{n\to\infty}f(x+b_n). \] This notion is so weak that it is impossible to find a function $f\colon\real\to\real$ which is nowhere weakly continuous. This follows from the following easy, but a little surprising theorem. \medskip \begin{theorem}[{\cite[p. 82]{CollingwoodLohwater:ThClSets}}]%2.16 \label{th:WeakC} Every function $f\colon\real\to\real$ is weakly continuous everywhere on the complement of a countable set. \end{theorem} \medskip A natural symmetric counterpart of weak continuity is defined as follows. A function $f\colon\real\to\real$ is {\em weakly symmetrically continuous} at $x$ if there is a sequence $h_n\to 0$ such that \[ \lim_{n\to\infty}(f(x+h_n)-f(x-h_n))=0. \] However, the symmetric version of Theorem~\ref{th:WeakC} badly fails: there exist nowhere weakly symmetrically continuous functions (which are also called {\em uniformly antisymmetric functions}). Their existence follows immediately from the following theorem of Ciesielski and Larson from~1993. \medskip \begin{theorem}[Ciesielski, Larson \cite{CL}]%2.17 \label{th:CLUnifA} There exists a function \linebreak $f\colon\R\to\natural$ such that the set \[ S_x=\{h>0\colon f(x+h)=f(x-h)\} \] is finite for every $x\in\R$. \end{theorem} \medskip The function $f$ from Theorem~\ref{th:CLUnifA} raises the questions in two directions. Can the range of $f$ be any smaller? Can the size of all sets $S_x$ be uniformly bounded? The first of this questions leads to the following open problem from~\cite{CL}. (See also problems listed in~\cite{Th}.) \begin{problem} %2 \label{pr:CLUA} Does there exist a uniformly antisymmetric function \linebreak $f\colon\real\to\R$ with range $f[\R]$ being (a) finite? (b) bounded? \footnote{A uniformly antisymmetric function $f\colon \real \to [0,1]$ has been recently constructed by S. Shelah.} \end{problem} \medskip Concerning part (a) of this problem it has been proved in~1993 by Ciesielski~\cite{C0} that the range of a uniformly antisymmetric function must have at least $4$ elements. (Compare also~\cite{C4}.) The estimation of sizes of sets $S_x$ from Theorem~\ref{th:CLUnifA} has been examined by Komj\'{a}th and Shelah in~1993, leading to the following two theorems. \medskip \begin{theorem}[Komj\'{a}th, Shelah \cite{KomSh}]%2.18 \label{th:KomSh1}The Continuum Hypothesis is equivalent to the existence of a function $f\colon\R\to\natural$ such that the set \[ S_x=\{h>0\colon f(x+h)=f(x-h)\} \] has at most $1$ element for every $x\in\R$. \end{theorem} \medskip \begin{theorem}[Komj\'{a}th, Shelah \cite{KomSh}]%2.19 \label{th:KomSh2} If $\continuum>\omega_{k+1}$, $k=0,1,2,\ldots$, then there is no function $f\colon\R\to\natural$ such that the set \[ S_x=\{h>0\colon f(x+h)=f(x-h)\} \] has at most $2^k$ elements for every $x\in\R$. \end{theorem} \medskip Theorem~\ref{th:KomSh1} suggests that the converse of Theorem~\ref{th:KomSh2} should also be true. However, this is still unknown, leading to another open problem. \medskip \begin{problem}%3 \label{pr:SandD} Does the assumption that $\continuum\leq\omega_{k+1}$ imply that there exists a function $f\colon\R\to\natural$ such that the set \[ S_x=\{h>0\colon f(x+h)=f(x-h)\} \] has at most $2^k$ elements for every $x\in\R$? \end{problem} \medskip For $k=0$ the positive answer is implied by Theorem~\ref{th:KomSh1}. Also, it is consistent that $\continuum=\omega_{k+1}$ and there exists $f\colon\R\to\natural$ such that each $S_x$ has at most $2^k$ elements. This follows from another theorem of Komj\'{a}th and Shelah \cite[Thm 1]{KomSh2}. (See also a paper~\cite{C:SumsAndDiff} of Ciesielski related to this subject.) In fact, the proof of Theorem~\ref{th:CLUnifA} gives also the following version for functions on $\real^n$: \begin{itemize} \item There exists a function $f\colon\R^n\to\natural$ such that the set \[ \{h\in\real^n\colon f(x+h)=f(x-h)\} \] is finite for every $x\in\R^n$. \end{itemize} This statement is related to the following recent theorem of J.~Schmerl, which solves a long standing problem of Erd\H{o}s~\cite[Problem~15.9]{millerq}. (See also a survey article~\cite{Kom} for more on this problem.) \medskip \begin{theorem}[Schmerl~\cite{Schmerl}]%2.20 \label{th:Schmerl} There exists a function $f\colon\R^n\to\natural$ such that for any distinct $a,b,x\in\real^n$ with $\|a-x\|=\|x-b\|$ all the values $f(a)$, $f(x)$ and $f(b)$ are not equal. \end{theorem} \medskip \noindent Thus, this theorem says, that there exists (in ZFC) a countable partition of $\real^n$ such that no three vertices $a,b,x$ spanning isosceles triangle belong to the same element of the partition. \medskip \stepcounter{section} \section*{\bf 3. New classic-like results} \medskip Consider a function $F=\la f_1,f_2\ra$ from $\real$ onto $\real^2$. By a well known theorem of Peano from~1890 (see e.g.~\cite{Sagan}) such an $F$ can be continuous. However, it is not difficult to see that it cannot be differentiable. It follows easily from the fact that every differentiable function $f\colon\real\to\real$ satisfies the Banach condition $T_2$, i.e., the set $\{y\colon f^{-1}(y)\text{ is uncountable}\}$ has Lebesgue measure zero. (See e.g. \cite[Chap. VII, p. 221]{Saks}.) Thus, Morayne in 1987 considered the following question: can function $F=\la f_1,f_2\ra$ be chosen in such a way that at every point $x\in\real$ either $f_1$ or $f_2$ is differentiable? The surprising answer is given below. \medskip \begin{theorem} [Morayne~\cite{Morayne}]%3.1 \label{th:Morayne} The Continuum Hypothesis is equivalent to the existence of a function $F=\la f_1,f_2\ra$ from $\real$ onto $\real^2$ such that at every point $x\in\real$ either $f_1$ or $f_2$ is differentiable. \end{theorem} \medskip The proof of this theorem is based on a well known theorem of Sierpi\'nski \cite[Property $P_1$]{Si:CH} from~1919 that CH is equivalent to the existence of a decomposition of $\real^2$ into two sets $A$ and $B$ such that all horizontal sections of $A$ and all vertical sections of $B$ are at most countable. It is also worthwhile to point out that the function $F$ from Theorem~\ref{th:Morayne} is not a Peano curve, since it is not continuous. In fact Morayne proves in the same paper that for such an $F=\la f_1,f_2\ra$ it is impossible that even one of $f_1$ or $f_2$ is measurable. \medskip Next, recall that if two continuous functions $f,g\colon\real\to\real$ agree on some dense set $M\subset\real$ then they are equal. Does the statement remain true if the clause ``agree on $M$'' is replaced by ``$f[M]=g[M]$?'' Clearly not, as shown by $M=\rational$ and any two different rational translations of the identity function. What about finding some more complicated set $M\subset\real$ for which the implication \[ \text{if $f[M]=g[M]$ then $f=g$} \] holds for any continuous $f$ and $g$? Even this is too much to ask, as recently noted by Burke and Ciesielski \cite[Remark 6.6]{BC}. On the other hand, the following theorem of Berarducci and Dikranjan from~1993 gives a positive (consistent) answer to this question in the class of continuous nowhere constant functions. (A function is nowhere constant if it is not constant on any non-empty open set.) \medskip \begin{theorem}[Berarducci, Dikranjan \cite{BD}]%3.2 \label{th:BerDikra} If the Con\-ti\-nu\-um Hy\-po\-the\-sis holds then there exists a set $M\subset\real$ (called {\rm magic}) such that for every continuous nowhere constant functions $f,g\colon\real\to\real$, \[ \text{if $f[M]\subset g[M]$ then $f=g$.} \] \end{theorem} \medskip The construction of a magic set given in~\cite{BD} is done by an easy diagonal transfinite induction argument and uses only the assumption that less than continuum many meager sets do not cover $\real$. In particular, CH can be replaced by MA in Theorem~\ref{th:BerDikra}. Examining the problem of existence of a magic set in ZFC Burke and Ciesielski noticed the following properties of a magic set. \medskip \begin{theorem}[Burke, Ciesielski \cite{BC}]%3.3 \label{th:CBurke1} If $M\subset\real$ is a magic set then \begin{itemize} \item[(a)] $M$ is dense and nowhere meager; \item[(b)] $f[M]\not\supset[0,1]$ for every continuous $f\colon\R\to\R$. \end{itemize} \end{theorem} \medskip In fact part (b) of Theorem~\ref{th:CBurke1} is just a remark: if there were a continuous $f\colon\R\to\R$ with $f[M]\supset[0,1]$ then it could be easily modified to a nowhere constant function such that $f[M]=\real$, and the functions $f$ and $g=1+f$ would give a contradiction. But (b) shows that there is no magic set of cardinality continuum in the model from Theorem~\ref{th:sierp2NEG}, the iterated perfect set model. Although it was noticed in \cite{BC} that in this model there exists a magic set (clearly of cardinality less than $\continuum$), Theorem~\ref{th:CBurke1} was used by Ciesielski and Shelah as a base in proving that magic set cannot be constructed in ZFC. \medskip \begin{theorem}[Ciesielski, Shelah \cite{CShel}]%3.4 \label{th:CShelah} There is a model of ZFC in which \begin{itemize} \item[(a)] every subset of\/ $\R$ of cardinality less than $\continuum$ is meager; \item[(b)] for every set $M\subset\R$ of cardinality continuum there exists a continuous function $f\colon\R\to\R$ such that $f[M]=[0,1]$. \end{itemize} In particular, there is no magic set in this model. \end{theorem} \medskip The magic sets for different classes of functions have also been considered. Burke and Ciesielski~\cite{BC} studied such sets (which they call {\em sets of range uniqueness}\/) for the classes of measurable functions with respect to abstract measurable spaces with negligibles. In particular, they proved the following theorem concerning the Lebesgue measurable functions. \newpage %\medskip \begin{theorem}[Burke, Ciesielski \cite{BC}]%3.5 \label{th:CBurke2} \ \begin{itemize} \item[(a)] If $\cov(\calN)=\continuum$ (thus under CH or MA) then there exists a set $M\subset\R$ with the property that for every measurable functions $f,g\colon\R\to\R$ which are not constant on any set of positive measure \[ \text{if $f[M]\subset g[M]$ then $f=g$ almost everywhere.} \] \item[(b)] There is a model of ZFC in which a set from part (a) does not exist. \end{itemize} \end{theorem} \medskip The model satisfying Theorem~\ref{th:CBurke2}(b) is a modification of the iterated perfect set model and was constructed by Corazza \cite{Corazza} in~1989. Once again it satisfies property (b) of Theorem~\ref{th:CShelah}, while part (a) is replaced by $\cov(\calN)=\continuum$. It has also been proved by Burke, Ciesielski, and Larson that for the class $D^1$ of differentiable functions the existence of a magic set can be proved in ZFC. \medskip \begin{theorem}[Burke, Ciesielski \cite{CL:magic}]%3.6 \label{th:CLmagic} There exists a set $M\subset\R$ such that for every $D^1$ nowhere constant functions $f,g\colon\R\to\R$ \[ \text{if $f[M]\subset g[M]$ then $f=g$.} \] \end{theorem} \medskip \noindent Note also that the existence of a countable magic set (a convergent sequence) for the class of analytic functions has been proved already in~1981 by Diamond, Pomerance, and Rubel~\cite{DPR}. However, not all convergent sequences form a magic set for this class. \medskip For the following consideration recall that a function $f\colon\R^n\to\R$ is {\em Darboux} (or {\em has the Darboux property}\/) if $f[C]$ is connected for every connected subset $C$ of $\R^n$. Thus, in case of $n=1$ Darboux functions are precisely the functions for which the Intermediate Value Theorem holds. The class of Darboux functions will be denoted here by $\D$ (with $n$ clear from the context, usually $n=1$). The class of Darboux functions has been studied for a long time as one of possible generalizations of the class of continuous functions. (Clearly every continuous function is Darboux.) However, it has some peculiar properties. For example, it is not closed under addition. In fact, in 1927 Lindenbaum \cite{Li} noticed (without a proof) that every function $f\colon\real\to\real$ can be written as a sum of two Darboux functions. (For proofs, see \cite{Si3, SMa}.) This theorem has been improved in several ways. Erd\H{o}s~\cite{Erdos} showed that if $f$ is measurable, both of the summands can be chosen to be measurable. Another improvement was done by Fast~\cite{F} in~1959 who proved that for every family $\F$ of real functions that has cardinality continuum there is just one Darboux function $g$ such that the sum of $g$ with any function in $\calF$ has the Darboux property. The natural question of whether such a ``universal'' summand exists also for families of larger cardinality has been studied by Natkaniec~\cite{N1} and lead to the development described in Section~4. %\ref{sec:CardF}. A problem that is in some sense opposite to the existence of a ``universal'' summand is for which families $\calF$ of functions there is a ``universally bad'' Darboux function $g$, in the sense that the sum of $g$ with any function in ${\calF}$ does not have the Darboux property. In~1990 Kirchheim and Natkaniec addressed this problem for the class $\F$ of continuous nowhere constant functions. \medskip \begin{theorem}[Kirchheim, Natkaniec~\cite{NK}] %3.7 \label{th:NatKir} If union of less than $\continuum$ many meager subsets of\/ $\real$ is meager (thus under CH or MA) then there exists a Darboux function $g\colon\R\to\R$ such that $f+g$ is not Darboux for every continuous nowhere constant function $f\colon\R\to\R$. \end{theorem} \medskip The problem whether the additional set-theoretic assumptions are necessary in this theorem was investigated in~1992 by Komj\'ath~\cite{Km} and was settled in~1995 by Stepr\=ans. \medskip \begin{theorem}[Stepr\=ans~\cite{St}]%3.8 \label{th:Stepr1} It is consistent with ZFC that for every Darboux function $g\colon\R\to\R$ there exists a continuous nowhere constant function $f\colon\R\to\R$ such that $f+g$ is Darboux. \end{theorem} \medskip A model having this property is the iterated perfect set model. Note also that in Theorem~\ref{th:NatKir} the restriction to the nowhere constant functions is important. This has been proved independently by T.~Natkaniec (in his 1992$/$93 paper~\cite{Na2a}) and by J.~Stepr\=ans (in the~1995 paper mentioned above). \medskip \begin{theorem}[Natkaniec~\cite{Na2a}, Stepr\=ans~\cite{St}]%3.9 \label{th:Stepr2} For every Dar\-bo\-ux \linebreak func\-tion $g\colon\R\to\R$ there exists a continuous non-constant function $f\colon\R\to\R$ such that $f+g$ is Darboux. \end{theorem} \medskip To state further results recall the following generalizations of continuity. A function $f\colon\R^n\to\R$ is {\em almost continuous} (in the sense of Stallings) if each open subset of $\real^n\times\real$ containing the graph of $f$ contains also a continuous function from $\R^n$ to $\R$~\cite{Stal}. Function $f\colon\R\to\R$ {\it has a perfect road\/} at $x\in\R$ if there exists a perfect set $C$ such that $x$ is a bilateral limit point of $C$ and $f\restriction C$ is continuous at $x$~\cite{Max}. The classes of all almost continuous functions and all functions having a perfect road at each point are denoted by $\op{AC}$ and $\op{PR}$, respectively. It is easy to see that $\C\subset\op{AC}\subset\D$ (for functions on $\real$) and that the inclusions are strict (see e.g.~\cite{BHL}), where $\C$ stands for the class of all continuous functions. We will also consider the class $\op{SZ}$ of {\em Sierpi\'nski-Zygmund (SZ-) functions}, i.e., functions $f\colon\real\to\real$ whose restrictions $f\restriction X$ are discontinuous for all subsets $X$ of $\R$ of cardinality continuum. (That is, functions from Theorem~\ref{th:SZ}.) The classes $\op{SZ}$ and $\op{PR}$ recently appeared in a~1993 paper of Darji \cite{D}, who constructed in ZFC a function $f\in\op{SZ}\cap\op{PR}$. Answering a question posed by Darji the following theorem has been proved recently by Balcerzak, Ciesielski, and Natkaniec. \medskip \begin{theorem}[Balcerzak, Ciesielski, Natkaniec \cite{BCN}]%3.10 \label{th:BCN} \ \begin{itemize} \item[(a)] If\/ $\R$ is not a union of less than continuum many of its meager subsets (thus under CH or MA) then there exists an $f\in\op{SZ}\cap\op{PR}\cap\op{AC}$. \item[(b)] There is a model of ZFC in which every Darboux function $f\colon\real\to\real$ is continuous on some set of cardinality continuum. In particular, in this model we have $\op{SZ}\cap\op{AC}=\op{SZ}\cap\D=\emptyset$. \end{itemize} \end{theorem} \medskip The model satisfying Theorem~\ref{th:BCN}(b) is, once again, the iterated perfect set model. \medskip Another generalization of continuity is that of countable continuity: a function $f\colon\R\to\R$ is {\em is countably continuous} if there exists a countable partition $\left\{X_n\right\}_{n=1}^\infty$ of $\R $ such that the restriction of $f$ to any $X_n$ is continuous. (See also Section~4.) %\ref{sec:CardF}.) In~1995 Darji gave the following combinatorial characterization of this notion. \medskip \begin{theorem}[Darji \cite{D1,D2}]%3.11 \label{th:DarjCountCon} If the Con\-ti\-nu\-um Hy\-po\-the\-sis holds \linebreak then \begin{itemize} \item[($\star$)] $f\colon\R\to\R$ is countably continuous if and only if for every uncountable set $U\subseteq\R $ there is an uncountable set $V\subseteq U$ such that the restriction $f\restriction V$ is continuous. \end{itemize} \end{theorem} \medskip The characterization ($\star$) cannot be proved in ZFC. This follows from a result of Cicho\'n and Morayne~\cite{CiMo} from 1988 which implies that in some models of ZFC (actually, when $\continuum=\omega_2$ and $d=\omega_1$, where $d$ is the dominating number) ($\star$) is false. However, it is not known, whether the equivalence ($\star$) can be proved in absence of CH, leading to the following open problem. \medskip \begin{problem}%4 \label{pr:Darj} {\em Is ($\star$) from Theorem~\ref{th:DarjCountCon} equivalent to the Continuum Hypothesis?} \end{problem} \medskip Another recent theorem concerning countable and symmetric continuities is the following theorem of Ciesielski and Szyszkowski, answering a question of L.~Larson. \medskip \begin{theorem}[Ciesielski, Szyszkowski \cite{CSz}]%3.12 \label{th:CSz} There exists a sym\-me\-tri\-ca\-lly continuous function $f\colon\R\to\R$ such that for some set $Z\subset\real$ of cardinality continuum $f\restriction Z$ is of Sierpi\'nski-Zygmund type, i.e., $f\restriction X$ is discontinuous for any subset $X$ of $Z$ of cardinality continuum. In particular, $f$ is not countably continuous. \end{theorem} \medskip We will finish this section with the following two interesting results. The first one was proved independently in~1978 by Grande and Lipi\'nski and in~1979 by Kharazishvili. \medskip \begin{theorem}[Grande,\ Lipi\'nski~\cite{GrLp},\ Kharazishvili~\cite{Kh}] %3.13 \label{th:GLandK} \ If \ the \linebreak Con\-ti\-nu\-um Hypothesis holds then there exists a non-measurable function $F\colon\R^2\rightarrow\R$ such that for every measurable $f\colon\R \rightarrow\R$, the composition $F(x,f(x))$ is measurable. \end{theorem} \medskip This theorem has important consequences concerning the existence of solutions of the differential equation $y^{\prime }=F(x,y)$ in the class of absolutely continuous functions. In~1992 Balcerzak~\cite{B1} showed that in Theorem~\ref{th:GLandK} the CH assumption can be weakened to $\non(\N)=\continuum$. However, the following problem remains open. \medskip \begin{problem}%5 \label{Balc} {\em Can Theorem~\ref{th:GLandK} be proved in ZFC?} \end{problem} \medskip \noindent In fact, all functions $F$ satysfying Theorem~\ref{th:GLandK} are of the form $\charf{h}$, where $h$ is a (partial) function from $\real $ to $\real $. It is worth to mention here that, by~\cite[Prop.~1.5]{B1}, the property considered in Problem~\ref{pr:Blum} implies that no $F=\charf{h}$ with $h\colon\real\to\real$ can satisfy Theorem~\ref{th:GLandK}. Similarly, the property considered in the following stronger version of Problem~\ref{pr:Blum} implies that $F$ from Theorem~\ref{th:GLandK} cannot be of the form $\charf{h}$ for a function $h$ from $Y\subset\real$ into $\real$. \medskip \begin{problem}%6 \label{BalcGen} Is it consistent that for every subset $Y$ of $\real$ of positive outer measure and every function $f\colon Y\to\real$ there exists a set $X\subset Y$ of positive outer measure such that $f\restriction Y$ is continuous? \end{problem} \medskip The second result is the following 1974 theorem of R.~O.~Davies. \medskip \begin{theorem}[Davies~\cite{Davies}]%3.14 \label{th:Davies} If the Con\-ti\-nu\-um Hypothesis holds then for every $f\colon\R^2\to\R$ there exist functions $g_n,h_n\colon\R\to\R$, $n<\omega$, such that \[ f(x,y)=\sum_{n=0}^\infty g_n(x)\cdot h_n(y). \] \end{theorem} \medskip \begin{problem}[{\cite[Problem~15.11]{millerq}}]%7 \label{pr:Davies} {\em Is the con\-clu\-sion of Theorem~\ref{th:Davies} \linebreak e\-qu\-i\-va\-lent to CH?} \end{problem} \medskip Note that Theorem~\ref{th:Davies} is related to Hilbert's Problem~13 (from his famous Paris lecture of~1900) and a~1957 theorem of Kolmogorov, in which he proves that every continuous function $f\colon[0,1]^n\to\real$ can be represented in a certain form (similar to the above) by continuous functions of one variable. An interesting account on this and related results can be found in a~1984 paper of Sprecher~\cite{Sprec}. %\input{survey2} \medskip \stepcounter{section} \section*{\bf 4. Cardinal functions in analysis} \label{sec:CardF} \medskip The important recent developments in set theoretical analysis concern the cardinal functions that are defined for different classes of real functions. These investigations seem to be analogous to those concerning of cardinal functions in topology from the 1970's and 1980's. (See \cite{Ju2,Ho,Ju3,V}.) They are also related to the deep studies of cardinal invariants associated with different small subsets of the real line. (For a summary of the results concerning cardinals related to the measure and category see~\cite{Fr} or \cite{BJ}. For a survey concerning cardinals associated with the thin sets derived from harmonic analysis see~\cite{BKR}.) The first group of functions is motivated by the notion of countable continuity and was introduced in~1991 by J.~Cicho\'{n}, M.~Morayne, J.~Pawlikowski, and S.~Solecki in~\cite{cimopaso}. More precisely, they define the {\em decomposition} function $\dec(\F,\G)$ for arbitrary families $\F\subset\real^\real$ and $\G\subset\bigcup\{\real^X\colon X\subset\real\}$, where $Y^X$ stands for the set of all functions from $X$ to $Y$. \[ \dec(\F,\G)=\min\{\kappa\leq\co\colon(\forall f\in\F) (\exists {\cal X}\in\Pi_\kappa) (\forall X\in{\cal X})(f\restriction X\in\G)\}\cup\{\co^+\}, \] where $\Pi_\kappa$ denotes the family of all coverings of $\real$ with at most $\kappa$ many sets. In particular, if $\C$ stands for the family of all continuous functions (from subsets of $\real$ into $\real$) then \[ \text{$f\colon\real\to\real$ is countably continuous if and only if $\dec(\{f\},\C)\leq\omega$.} \] In~\cite{cimopaso} the authors considered the values of $\dec(\B_\beta,\B_\alpha)$ for $\alpha<\beta<\omega_1$, where $\B_\alpha$ stands for the functions of $\alpha$-th Baire class. The motivation for this definition comes from a question of N.~N.~Luzin whether every Borel function is countable continuous. This question was answered negatively by P.~S.~Novikov (see Keldy\v{s}~\cite{Keld}) and was subsequently generalized by Keldy\v{s}~\cite{Keld} (in~1934), and S.~I.~Adian and P.~S.~Novikov~\cite{AdiNov} (in~1958). The most general result in this direction was obtained in late 1980's by M.~Laczkovich (see Cicho\'{n}, Morayne~\cite{CiMo}) who proved, in particular, that $\dec(\B_\beta,\B_\alpha)>\omega$ for every $\alpha<\beta<\omega_1$. One of the most interesting results from the paper~\cite{cimopaso} is the following theorem. \medskip \begin{theorem}[Cicho\'{n}, Morayne, Pawlikowski, Solecki~\cite{cimopaso}]%4.1 \label{th:CMPS} \[ {\rm cov}(\M)\leq \dec(\B_1,\C)\leq d, \] where ${\rm cov}(\M)$ is the smallest cardinality of a covering of $\real$ by meager sets, and $d$, the dominating number, is the smallest cardinality of a dominating family $D\subset\omega^\omega$, i.e., such that for every $f\in\omega^\omega$ there exists $g\in D$ with $f\leq^* g$.% \footnote{Recall that for $f,g\in\omega^\omega$ we write $f\leq^* g$ if there exists an $n<\omega$ such that $f(m)\leq g(m)$ for every $m>n$.} \end{theorem} \medskip \noindent It has been also shown by J.~Stepr\={a}ns and S.~Shelah that none of these inequalities can be replaced by the equation. \medskip \begin{theorem}[Stepr\={a}ns~\cite{step.30}]%4.2 \label{th:Stdec} It is consistent with ZFC that \[ {\rm cov}(\M)<\dec(\B_1,\C). \] \end{theorem} \medskip \begin{theorem}[Shelah, Stepr\={a}ns~\cite{step.34}]%4.3 \label{th:ShStdec} It is consistent with ZFC that \[ \dec(\B_1,\C)\omega$ there exists a model of ZFC in which $\co=\kappa$ and $\dec(\op{SZ},\C)=\co$. \item[(2)] For every $\kappa$ with $\cf(\kappa)>\omega$ there exists a model of ZFC in which $\co=\kappa$ and $\dec(\op{SZ},\C)=\cf(\kappa)=\cf(\co)$. \end{itemize} \end{theorem} \medskip In fact, (1) happens in a model obtained by extending a ground model with GCH by adding $\kappa$ many Cohen reals. The equation $\dec(\op{SZ},\C)=\co$ follows immediately from Theorem~\ref{th:Sh1}. The model for (2) is obtained as follows. You start with a model with GCH, assume that $\lambda={\rm cf}(\kappa)<\kappa$ and take an increasing sequence \linebreak $\{\lambda_\xi\colon\xi<\lambda\}$ cofinal with $\lambda$ and such that each $\lambda_\xi$ is a cardinal successor. The desired model is obtained by a generic extension via forcing $P$ which a finite support iteration of forcings $M_\xi$, where each $M_\xi$ is a standard ccc forcing adding the Martin's Axiom over the previous model and making $\co=\lambda_\xi$. \medskip The second group of cardinal functions is defined in terms of algebraic operations on functions. Their definition was motivated by the following property of Darboux functions (from $\R$ to $\R$) due to Fast and mentioned in the previous section: \begin{itemize} \item for every family ${\cal H}\subset\real^\real$ with $|{\cal H}|\leq\continuum$ \begin{equation} \label{con1} \text{ there exists $g\in\real^\real$ such that $g+h\in\D$ for every $h\in {\cal H}$, } \end{equation} \end{itemize} where $|Z|$ denotes the cardinality of $Z$. In~1974 Kellum~\cite{Ke} proved the similar result for the class $\op{AC}$ of almost continuous functions and in~1991 Natkaniec~\cite{N1} defined the following cardinal functions for every ${\calF}\subseteq\R^{\R}$ to study these phenomena more closely. \begin{align*} \op{A}({\calF}) &= %& \min\left\{\left|{\cal H}\right|\colon{\cal H}\subset\real^\real\ \text {\&}\ \neg\exists g\in\R^{\R }\ \forall h\in{\mathcal{H}}\ g+h\in{\calF}\right\} \cup\{(2^\continuum)^+\}\\ &= %& \min\left\{\left|{\mathcal{H}}\right|\colon{\mathcal{H}}\subset\real^\real\ \text{\&}\ \forall g\in\R^{\R }\ \exists h\in{\mathcal{H}}\ g+h\notin{\calF}\right\} \cup\{(2^\continuum)^+\}\\ \op{M}({\calF}) \!\! &= %& \!\! \min\left\{\left|{\mathcal{H}}\right|\colon{\mathcal{H}}\subset\real^\real\ \text{\&}\ \neg\exists g\in\R^{\R}\setminus\{\charf\emptyset\}\ \forall h\in{\mathcal{H}}\ g\cdot h\in{\calF}\right\} \cup\{(2^\continuum)^+\}\\ \!\! & =%& \!\! \min\left\{\left|{\mathcal{H}}\right|\colon{\mathcal{H}}\subset\real^\real\ \text{\&}\ \forall g\in\R^{\R}\setminus\{\charf\emptyset\}\ \exists h\in{\mathcal{H}}\ g\cdot h\notin{\calF}\right\} \cup\{(2^\continuum)^+\}. \end{align*}\label{defMUL} The extra assumption that $g\neq\charf\emptyset$ is added in the definition of $\op{M}$ since otherwise for every family $\F\su\real^\real$ containing constant zero function $\charf\emptyset$ we would have $\op{M}(\F)=(2^\co)^+$. Its easy to see that the functions $\op{A}$ and $\op{M}$ are monotone in a sense that $\op{A}(\F)\leq\op{A}(\G)$ and $\op{M}(\F)\leq\op{M}(\G)$ for every $\F\subset\G\subset\R^\R$. Also clearly (\ref{con1}) is false for ${\mathcal{H}}=\real^\real$. Thus, in language of the function $\op A$ the results of Fast and Kellum can be expressed as follows: \[ \continuum<\op{A}(\op{AC})\leq\op{A}(\D)\leq 2^\continuum. \] If $2^\continuum=\continuum^+$ (so, under the Generalized Continuum Hypothesis GCH) the values of $\op{A}(\op{AC})$ and $\op{A}(\D)$ are clear: $\op{A}(\op{AC})=\op{A}(\D)=2^\continuum=\continuum^+$. Thus, Natkaniec asked~\cite[p. 495]{N1} (see also \cite[Problem 1]{GMN}) whether the equation $\op{A}(\op{AC})=2^\continuum$ can be proved in ZFC. This question was investigated by Ciesielski and Miller in~1994. They proved that $\op{A}(\op{AC})=\op{A}(\D)$, that the cofinality $\op{cf}(\op{A}(\D))$ of $\op{A}(\D)$ is greater than $\continuum$, and that this together with the inequalities $\continuum<\op{A}(\D)\leq 2^\continuum$ is essentially all that can be proved in ZFC. \smallskip \begin{theorem}[Ciesielski, Miller \cite{CMi}]%4.7 \label{th:CMill1} \ \begin{itemize} \item[(a)] $\op{A}(\op{AC})=\op{A}(\D)={\frak e}_\continuum$, where \[ {\frak e}_\kappa=\min\{|F|\colon F\su \kappa^\kappa\ \text{\em \&}\ \forall g\in \kappa^\kappa\ \exists f\in F\ |f\cap g|<\kappa\}. \] \item[(b)] $\op{cf}(\op{A}(\D))>\continuum$. \item[(c)] Let $\lambda\geq\kappa\geq\omega_2$ be cardinals such that $\op{cf}(\lambda)>\omega_1$ and $\kappa$ is regular. Then it is relatively consistent with ZFC that the Continuum Hypothesis is true, $2^{\continuum}=\lambda$, and $\op{A}(\D)=\kappa$. \item[(d)] Let $\lambda$ be a cardinal such that $\op{cf}(\lambda)>\omega_1$. Then it is relatively consistent with ZFC that the Continuum Hypothesis holds and $\op{A}(\D)=\lambda=2^{\continuum}$. \end{itemize} \end{theorem} \smallskip \noindent In particular Theorem~\ref{th:CMill1} says that $\op{A}(\D)$ does not have to be a regular cardinal (part (d)) and that $\op{A}(\D)$ can be any regular cardinal number between $c^+$ and $2^c$, with $2^{c}$ being ``arbitrarily large'' (part (c)). At the same time Natkaniec and Rec\l aw established the values of $\op{M}(\op{AC})$ and $\op{M}(\D)$ proving %\smallskip \begin{theorem} [Natkaniec, Rec\l aw \cite{NR}]%4.8 \label{th:NR} $\op{M}(\op{AC})=\op{M}(\D)=\cf(\co)$. \end{theorem} \smallskip The first systematic study of functions $\op{A}$ and $\op{M}$ was done by Ciesielski and Rec\l aw in the later part of~1995. They collected basic properties of operators $\op{A}$ and $\op{M}$, which are stated below, and found the values of $\op{A}$ and $\op{M}$ for some other classes of functions. %\smallskip \begin{proposition}[\cite{CR}] %4.9 \label{prop:CRec} Let $\emptyset\neq\F\su\G\su\reals^\reals$. Then \begin{itemize} \item[(0)] $\add(\emptyset)=\mul(\emptyset)=1$; \item[(1)] $\add(\F)\leq\add(\G)$ and $\mul(\F)\le\mul(\G)$; \item[(2)] $\add(\F)\geq 2$; \item[(2$^\prime$)] $\mul(\F)\geq 2$ if $\charf\emptyset,\charf\real\in\F$; \item[(3)] $\add(\F)=\left(2^\co\right)^+$ if and only if $\F=\real^\real$; \item[(3$^\prime$)] $\mul(\F)\leq\co$ if $r\charf{\{x\}}\not\in\F$ for every $r,x\in\real$, $r\neq 0$; \item[(4)] $\add(\F)=2$ if and only if $\F-\F=\{f_1-f_2\colon f_1,f_2\in\F\}\neq\real^\real$.% \footnote{In~\cite{CR} this was proved with the additional assumption that $\charf\emptyset\in\F$. This extra assumption was removed by F.~Jordan in~\cite{FJordan}.} \end{itemize} \end{proposition} \medskip \noindent In particular, (4) from Proposition~\ref{prop:CRec} shows that every function is a difference of two functions from a class $\F$ if and only if $\add(\F)>2$. To state the other results from~\cite{CR} recall the definitions the following classes of functions, where $X$ is an arbitrary topological space. \begin{itemize} \item[$\conn(X)$] of {\em connectivity functions} $f\colon X\to\R$, i.e., such that the graph of $f$ restricted to $C$ (that is $f\cap[C\times\R]$) is connected in $X\times\R$ for every connected subset $C$ of $X$. \item[$\ext(X)$] of {\em extendable functions} $f\colon X\to\R$, i.e., such that there exists a connectivity function $g\colon X\times[0,1]\to\R$ with $f(x)=g(x,0)$ for every $x\in X$. \item[$\pc(X)$] of {\em peripherally continuous functions} $f\colon X\to\R$, i.e., such that for every $x\in X$ and any pair $U\su X$ and $V\in\R$ of open neighborhoods of $x$ and $f(x)$, respectively, there exists an open neighborhood $W$ of $x$ with $\cl(W)\su U$ and $f[\bd(W)]\su V$, where $\cl(W)$ and $\bd(W)$ stand for the closure and the boundary of $W$, respectively. \end{itemize} We will write $\conn$, $\ext$ and $\pc$ in place of $\conn(X)$, $\ext(X)$, and $\pc(X)$ if $X=\real$. Notice also, that $f\in\pc$ if and only if $f$ is weakly continuous, as defined on page~\pageref{pageWC}. For the generalized continuity classes of functions (from $\real$ into $\real$) defined so far we have the following proper inclusions $\subset$, marked by arrows $\longrightarrow$. (See~\cite{BHL}.) \begin{picture}(300,100) \thicklines \put(10,40){\makebox(0,0){$\ext$}} \put(80,80){\makebox(0,0){$\op{AC}$}} \put(160,15){\makebox(0,0){$\op{PR}$} } \put(160,80){\makebox(0,0){$\conn$}} \put(240,80){\makebox(0,0){$\D$} } \put(320,40){\makebox(0,0){$\pc$}} \put(30,50){\vector(1,1){30}} \put(100,80){\vector(1,0){30}} \put(190,80){\vector(1,0){30}} \put(260,80){\vector(3,-2){40}} \put(30,35){\vector(4,-1){100}} \put(190,10){\vector(4,1){100}} \end{picture} \centerline{Chart 1.} \medskip In particular, inclusions $\op{AC}\subset\conn\subset\D$, monotonicity of $\op A$ and Theorem~\ref{th:CMill1}(a) imply that $\add(\conn)=\add(\op{AC})=\add(\D)$. Similarly, Theorem~\ref{th:NR} implies that $\mul(\conn)=\mul(\op{AC})=\mul(\D)$. The values of $\op A$ and $\op M$ for the remaining classes are as follows. %\newpage \medskip \begin{theorem}[Ciesielski, Rec\l aw \cite{CR}]%4.10 \label{th:CRec} \ \begin{itemize} \item[(1)] $\add(\ext)=\add(\op{PR})=\co^+$. \item[(2)] $\add(\pc)=2^\co$. \item[(3)] $\mul(\ext)=\mul(\op{PR})=2$. \item[(4)] $\mul(\pc)=\co$. \end{itemize} \end{theorem} \medskip \noindent Notice also that $\ext\subset\op{AC}\cap\op{PR}\subset\op{Conn}\cap\op{PR} \subset{\mathcal{D}}\cap\op{PR}\subset\op{PR}$. Thus, by monotonicity of $\op A$ and the above theorem we obtain the following corollary. \begin{corollary}%4.11 \label{cor:KCcap} \ \begin{itemize} \item[(1)] $\add(\op{AC}\cap\op{PR})=\add(\op{Conn}\cap\op{PR})= \add({\mathcal{D}}\cap\op{PR})=\co^+$; and, \item[(2)] $\mul(\op{AC}\cap\op{PR})=\mul(\op{Conn}\cap\op{PR})= \mul({\mathcal{D}}\cap\op{PR})=2$. \end{itemize} \end{corollary} \medskip The values of functions $\add$ and $\mul$ for the class $\op{SZ}$ has been studied by Ciesielski and Natkaniec. First they noticed that if the definition of $\mul$ from page~\pageref{defMUL} is used then trivially $\mul(\op{SZ})=1$, since for any function $h\in\real^\real$ with $\left|h^{-1}(0)\right|=\co$ we have $g\cdot h\notin\op{SZ}$ for every $g\in\real^\real$. Thus, they modified the definition of $\mul(\op{SZ})$ to \begin{align*} \op{M}(\op{SZ}) & = %& \!\! \min\left\{\left|{\mathcal{H}}\right|\colon{\mathcal{H}}\subseteq {\mathcal{R}}_0\ \text{\&}\ \neg\exists g\in\R^{\R}\ \forall h\in{\mathcal{H}}\ g\cdot h\in\op{SZ}\right\} \cup\{(2^\continuum)^+\}, \end{align*} where \[ {\mathcal{R}}_0=\left\{f\in\R^\R\colon \left|f^{-1}(0)\right|<\co\right\}. \] With this agreement in place they proved the following result. \medskip \begin{theorem}[Ciesielski, Natkaniec~\cite{CNat}]%4.12 \label{th:CNat1} \ \begin{itemize} \item[(a)] $\mul(\op{SZ})=\add(\op{SZ})={\op d}_\continuum$, where \[ {\op d}_\kappa=\min\{|F|\colon F\su \kappa^\kappa \&\ \forall g\in \kappa^\kappa\ \exists f\in F\ |f\cap g|=\kappa\}. \] \item[(b)] Let $\lambda\geq\kappa\geq\omega_2$ be cardinals such that $\op{cf}(\lambda)>\omega_1$ and $\kappa$ is regular. Then it is relatively consistent with ZFC that the Continuum Hypothesis is true, $2^{\continuum}=\lambda$, and $\add(\op{SZ})=\op{A}(\D)=\kappa$. \item[(c)] Let $\lambda>\omega_2$ be a cardinal such that $\op{cf}(\lambda)>\omega_1$. Then it is relatively consistent with ZFC that the Continuum Hypothesis holds, $2^{\continuum}=\lambda$, and $\add(\op{SZ})=\co^+<2^{\continuum}=\op{A}(\D)$. \end{itemize} \end{theorem} \medskip However, the following problems remain open. \medskip \begin{problem}[{\cite[Problems 2.13 and 2.17]{CNat}}]%8 \label{pr:SZ} \ \begin{itemize} \item[(a)] Is it consistent that $\add(\op{SZ})>\op{A}(\D)$? \item[(b)] Can $\add(\op{SZ})$ be a singular cardinal? \end{itemize} \end{problem} \medskip Another systematic study of the operator $\op A$ was done by F.~Jordan in~1996. In his study he examined the values of $\op{A}(\neg\F)$ where $\neg\F=\R^\R\setminus\F$ and classes $\F$ are chosen from those discussed above. Notice that $\op{A}(\neg\F)$ has the following very nice interpretation: \newpage \[ \text{$\op{A}(\neg\F)$ is the smallest cardinality of an ${\cal H}\subset\R^\R$ such that $\F-{\cal H}=\R^\R$,} \] where $\F-{\cal H}=\{f-h\colon f\in\F\ \&\ h\in{\cal H}\}$. To make this study non-trivial Jordan notes first that the value of $\op{A}(\F)$ does not determine the value of $\op{A}(\neg\F)$: \medskip \begin{theorem}[Jordan~\cite{FJordan}]%4.13 \label{th:Jor1} For every cardinal number $2\leq\lambda\leq 2^{\cuum}$ there exists $\F\subseteq\real^{\real}$ such that $\add(\F)=2$ and $\add(\neg\F)=\lambda$. In particular, there exist families $\G,\F\subseteq\real^{\real}$ such that $\add(\F)=\add(\G)$ and $\add(\neg\F)\neq\add(\neg\G)$. \end{theorem} \medskip This paper~\cite{FJordan} contains also the following results, where for a cardinal number $\kappa$ and functions $f,g\colon X\to Y$ we define $[X]^\kappa=\{Y\subset X\colon|Y|=\kappa\}$ and $[f=g]=\{x\in X\colon f(x)=g(x)\}$. \medskip \begin{theorem}[{Ciesielski \cite[Thm.~7]{FJordan}}]%4.14 \label{thm:three} $\add(\neg\op{PC})=\omega_{1}$. \end{theorem} \medskip \begin{theorem}[Jordan~\cite{FJordan}]%4.15 \label{th:Jor2} \ \begin{itemize} \item[(1)] $\add(\neg\op{PR})=\add(\neg\ext)=2^{\cuum}$. \item[(2)] $\add(\op{SZ})=d_{\cuum}\leq\add(\neg\D) \leq\add(\neg\conn)\leq\add(\neg\ac)\leq d_{\cuum}^{*}$, where \begin{multline*} d_{\kappa}^{*} =\min\{|F|\colon F\subseteq \kappa^{\kappa}\ \&\ (\forall G \in [\kappa^{\kappa}]^{\kappa})(\exists f\in F)(\forall g\in G)\\ (|[f=g]|=\kappa)\}. \end{multline*} \item[(3)] $\add(\ac)=\add(\conn)=\add(\D)= {\frak e}_{\cuum}\leq\add(\neg\op{SZ})\leq{\frak e}_{\cuum}^{*}$, where \begin{multline*} {\frak e}_{\kappa}^{*} =\min\{|F|\colon F\subseteq \kappa^{\kappa}\ \&\ (\forall G \in [\kappa^{\kappa}]^{\kappa})(\exists f \in F)(\forall g\in G)\\ (|[f=g]|<\kappa)\}. \end{multline*} \item[(4)] If $\cuum^{<\cuum}=\cuum$ then $\same_{\cuum}=\same_{\cuum}^{*}$% \footnote{This part was proved by K.~Ciesielski.} and $\diff_{\cuum}=\diff_{\cuum}^{*}$. In particular \[ \add(\op{SZ})=d_{\cuum}=\add(\neg\D) =\add(\neg\conn)=\add(\neg\ac)=d_{\cuum}^{*} \] and \[ \add(\D)=\add(\conn)=\add(\ac)=\diff_{\cuum}=\add(\neg\op{SZ})=\diff_{\cuum}^{*}. \] \item[(5)] If $\cuum^{<\cuum}=\cuum$ and $\cuum=\lambda^{+}$ for some cardinal $\lambda$ then $\same_{\cuum}\leq\diff_{\cuum}$. In particular \[ \add(\neg\D)=\add(\neg\ac)=\add(\op{SZ})=\same_{\cuum}\leq \diff_{\cuum}=\add(\D)=\add(\ac)=\add(\neg\op{SZ}). \] \end{itemize} \end{theorem} \medskip The importance of the extra assumptions in (4) and (5) of Theorem~\ref{th:Jor2} is not clear. In particular, the following problem is still open. \medskip \begin{problem}%9 \label{prob:one} When does either $\same_{\cuum}=\same_{\cuum}^{*}$ or $\diff_{\cuum}=\diff_{\cuum}^{*}$ hold? \end{problem} \medskip Note also that (4) and (5) of Theorem~\ref{th:Jor2}, and Theorem~\ref{th:CNat1} imply immediately the following corollary. \medskip \begin{corollary}[Jordan~\cite{FJordan}]%4.16 \label{cor:Jor} \ \begin{itemize} \item[(1)] Let $\lambda\geq\kappa\geq\omega_{2}$ be cardinals such that $\cf(\lambda)>\omega_{1}$ and $\kappa$ is regular. Then it is relatively consistent with ZFC+CH that $2^{\cuum}=\lambda$ and \[ \add(\neg\D)=\add(\neg\ac)=\add(\op{SZ})= \add(\D)=\add(\ac)=\add(\neg\op{SZ})=\kappa. \] \item[(2)] Let $\lambda>\omega_{2}$ be a cardinal such that $\cf(\lambda)>\omega_{1}$. Then it is relatively consistent with ZFC+CH that $2^{\cuum}=\lambda$, and \[ \add(\neg\D)=\add(\neg\ac)=\add(\op{SZ})=\cuum^{+}< 2^{\cuum}=\add(\D)=\add(\ac)=\add(\neg\op{SZ}). \] \end{itemize} \end{corollary} \medskip Finally, the following three classes of functions have been brought to this picture. \begin{itemize} \item[$\CIVP$] of functions $f\colon\R\to\R$ having the {\it Cantor Intermediate Value Property}, i.e., such that for every $x,y\in\R$ and for each Cantor set $K$ between $f(x)$ and $f(y)$ there is a Cantor set $C$ between $x$ and $y$ such that $f[C]\subset K$; \item[$\SCIVP$] of functions $f\colon\R\to\R$ having the {\it Strong Cantor Intermediate Value Property}, i.e., such that for every $x,y\in\R$ and for each Cantor set $K$ between $f(x)$ and $f(y)$ there is a Cantor set $C$ between $x$ and $y$ such that $f[C]\subset K$ and $f\restriction C$ is continuous; \item[$\WCIVP$] of functions $f\colon\R\to\R$ having the {\it Weak Cantor Intermediate Value Property}, that is, such that for every $x,y\in\R$ with \linebreak $f(x)\co$. In particular \[ \add(\op{SZ}\cap\neg\D\cap\CIVP)=\add(\F)=\add(\op{PR})=\co^+ \] for every $\F\subset\R^\R$ such that $\op{SZ}\cap\neg\D\cap\CIVP\subset\F\subset\op{PR}$. \end{theorem} \medskip \noindent This theorem gives the value of $\add$ for many classes that can be obtained intersecting classes from Chart~2 and $\op{SZ}$. Some of the difficulties of studying operators $\add$ and $\mul$ for the intersections $\op{AC}\cap\SCIVP$ and $\op{AC}\cap\CIVP$ is that there is relatively little known about these classes. For example, although Theorem~\ref{th:KBanNat}(2) implies that consistently these classes are different, a ZFC example was unknown until the following very recent theorem of Ciesielski. \medskip \begin{theorem} [Ciesielski~\cite{SomeDarbF}]%4.19 \label{th:KCnew} There exists an $f\in\op{AC}\cap\CIVP$ which is discontinuous on every perfect set. In particular, $f\notin\SCIVP$. \end{theorem} \medskip \noindent Also the only example of an $f\in\op{AC}\cap\SCIVP\setminus\ext$ results from the following recent theorem of Rosen. %(See also \cite{SomeDarbF} for a generalization.) \medskip \begin{theorem}[Rosen~\cite{Rosen}] %4.20 \label{th:ROSENnew} If the Continuum Hypothesis holds then there exists an $f\in\op{AC}\cap\SCIVP\setminus\ext$. \end{theorem} \medskip \noindent In fact, the conclusion of Theorem~\ref{th:ROSENnew} remains true under the assumption that union of less than continuum many meager sets is meager. However, the problem of existence of such a function in ZFC remains open. %\bibitem{Rosen} H.~Rosen, {\it %An almost continuous nonextendable function}, preprint. %\bibitem{SomeDarbF} K.~Ciesielski, {\it Some Darboux-like functions} Several other operators similar to $\add$ and $\mul$ have also been studied. Thus, in~1995 Natkaniec~\cite{N3} introduced the following operators connected to the composition of functions, where $\Const$ stands for the family of all constant functions. \begin{align*} \cout(\F) & \! = \! \min\! \left\{\left|{\mathcal{H}}\right|\!\colon{\mathcal{H}}\subset\real^\real\ \&\ \! \neg\exists g\in\R^{\R}\setminus\! \Const\ \forall h\in{\mathcal{H}}\ g\circ h\in{\calF}\right\} \!\cup\!\{(2^\continuum)^+\}\\ \cin(\F) &\! = \! \min \!\left\{\left|{\mathcal{H}}\right|\!\colon{\mathcal{H}}\subset\real^\real\ \&\ \! \neg\exists g\in\R^{\R}\setminus\! \Const\ \forall h\in{\mathcal{H}}\ h\circ g\in{\calF}\right\} \!\cup\!\{(2^\continuum)^+\} \end{align*} He proved also the following. \medskip \begin{theorem}[Natkaniec~\cite{N3}]%4.21 \label{th:NatComp} \ \begin{itemize} \item[(1)] $\cout(\ext)=\cout(\CIVP)=\cout(\op{PR})=1$. \item[(2)] $\cout(\ac)=\cout(\conn)=\cout(\D)=\cf(\co)$. \item[(3)] $\cout(\pc)=\co$. \item[(4)] $\cin(\ext)=\cin(\ac)=\cin(\conn)=\cin(\D)=1$. \item[(5)] $\cin(\CIVP)=\cin(\op{PR})=\cin(\pc)=\co^+$. \end{itemize} \end{theorem} \medskip Similar functions have been also studied by Ciesielski and Natkaniec~\cite{CNat}: \begin{align*} \cout(\op{SZ}) &\! = \! \min(\{\left|{\mathcal{H}}\right|\colon{\mathcal{H}}\subseteq{\mathcal{R}}_{out}\ \&\ \neg\exists g\in\R^\R\ \forall h\in\mathcal{H}\ g\circ h\in\op{SZ}\} \cup\{(2^{\co})^+\})\\ \cin(\op{SZ}) & \! = \! \min(\{\left|\mathcal{H}\right|\colon \mathcal{H}\subseteq{\mathcal{R}}_{in}\ \&\ \neg\exists g\in\R^\R\ \forall h\in\mathcal{H}\ h\circ g\in\op{SZ}\}\cup\{(2^{\co})^+\}) \end{align*} where ${\mathcal{R}}_{out}$ (${\mathcal{R}}_{in}$) is the set of all $h\in\R^\R$ for which there exists $g\in\R^\R$ such that $g\circ h\in\op{SZ}$ ($h\circ g\in\op{SZ}$, respectively). In fact, the classes ${\mathcal{R}}_{out}$ and ${\mathcal{R}}_{in}$ have the following nice characterizations: \[ {\mathcal{R}}_{out}=\left\{f\in\R^\R\colon \left|f^{-1}(y)\right|<\co\ \text{ for every } y\in\R\right\}, \] and, when $\co$ is a regular cardinal, \[ {\mathcal{R}}_{in}=\left\{f\in\R^\R\colon \left|f[\R]\right|=\co\right\}. \] %In fact, the class %${\mathcal{R}}_{out}$ has the following nice characterization: %\[{\mathcal{R}}_{out}=\left\{f\in\R^\R\colon \left|f^{-1}(y)\right|<\co\ % \text{ for every } y\in\R\right\}.\]\label{pageRout} In~\cite{CNat} the authors proved that \newpage %\medskip \begin{theorem}[Ciesielski, Natkaniec~\cite{CNat}] %4.22 \label{th:CNatComp} \ \begin{itemize} \item[(1)] $\cin(\op{SZ})=2$. \item[(2)] $\cout(\op{SZ})=\add(\op{SZ})$ if $\co=\lambda^+$ for some cardinal $\lambda$. \item[(3)] $\co<\cout(\op{SZ})\leq 2^\co$, if $\co$ is regular. \item[(4)] $\cf(\co)\leq\cout(\op{SZ})\leq 2^{\cf(\co)}$, if $\co$ is singular. \end{itemize} \end{theorem} \medskip Also, in a recent short survey paper~\cite{Nat:ShortSur} Natkaniec evaluated the values of operators $\add$, $\mul$, $\cin$ and $\cout$ for the class $\op{HAC}$ of {\em almost continuous functions in sense of Husain}, i.e., such $f\colon\R\to\R$ that $f^{-1}(U)\subset\op{int}(\cl(f^{-1}(U)))$ for every non-empty open set $U\subset\R$. \medskip \begin{theorem}[Natkaniec~\cite{Nat:ShortSur}]%4.23 \label{th:NatHus} \ \begin{itemize} \item[(1)] $\add(\op{HAC})=2^\co$. \item[(2)] $\mul(\op{HAC})=\cov(\M)$ , where $\cov(\M)$ is the smallest cardinality of a family of meager sets that covers $\R$. \item[(3)] $\cin(\op{HAC})=\left(2^\co\right)^+$ and $\cout(\op{HAC})=\co$. \end{itemize} \end{theorem} \medskip Some other cardinal operators connected with composition and concerning some kind of coding were also studied by Ciesielski and Rec{\l }aw~\cite{CR}, Ciesielski and Natkaniec~\cite{CNat}, and Natkaniec~\cite{Nat:ShortSur}. Another variant of function $\add$ is connected to the families of bounded functions. To define it properly the following notation is necessary. For a Let $\op{UB}$ stand for all uniformly bounded families ${\mathcal{H}}\subset\R^\R$, and let ${\mathcal{B}}$ be the class of all bounded functions $f\colon\R\to{\R}$. Then we define $$ \begin{array}{rl} \add_b({\calF})\!\!\! & =\min\left\{\left|{\mathcal{H}}\right|\colon {\mathcal{H}}\in\op{UB}\ \&\ \neg\left(\exists g\in{\mathcal{B}}\right) \left(\forall h\in{\mathcal{H}}\right)\left(g+h\in{\calF}\right)\right\}. \end{array} $$ In~1994 Maliszewski~\cite{Ma} proved that \[ \add_b({\mathcal{D}})=\cf(\co) \] so that $\add_b({\mathcal{D}})<\add({\mathcal{D}})$. Moreover, he proved that if ${\cal F}\in\op{UB}$, all functions in $\cal F$ are measurable (have Baire property), and the size of ${\cal F}$ is less than the additivity of measure (category) then there exists a ``universal summand'' bounded function for ${\cal F}$ with the same property. Similar results were also proved for countable families of Borel measurable functions of $\alpha$ class when $\alpha>1$ and for finite families of Baire one functions. The values of $\add_b$ for the other classes of functions from Chart~1 has been investigated by Ciesielski and Maliszewski~\cite{CMa}. In particular, they proved \newpage %\medskip \begin{theorem}[Ciesielski, Maliszewski~\cite{CMa}]%4.24 \label{th:CMal} \ \begin{itemize} \item[(1)] $\add_b(\ext)=\add_b(\op{PR})=2$. \item[(2)] $\add_b(\ac) =\add_b(\conn) =\add_b(\D)=\cf(\co)$. \item[(3)] $\add_b(\pc)=\co$. \end{itemize} \end{theorem} \medskip \noindent Notice also that Theorem~\ref{th:CMal} implies immediately the following corollary. \medskip \begin{corollary}[Ciesielski, Maliszewski~\cite{CMa}]%4.25 \label{cor:CMal} \ \begin{itemize} \item[(1)] Every bounded function $f\colon\reals\to\reals$ is the sum of two bounded almost continuous functions. \item[(2)] There exists a bounded function $f\colon\reals\to\reals$ which is not the sum of two bound\-ed functions with perfect road. \end{itemize} \end{corollary} \medskip \noindent In particular, Corollary~\ref{cor:CMal}(1) generalizes a result of Darji and Humke~\cite{DH} that every bounded function can be expressed a sum of three bounded almost continuous functions. On the other hand Corollary~\ref{cor:CMal}(2) shows that the following result of Natkaniec is sharp. \medskip \begin{theorem}[Natkaniec~\cite{Na2}] %4.26 \label{th:NatExt} Every bounded function can be expressed as a sum of three bounded extendable functions. \end{theorem} \medskip It might be also interesting to examine a bounded version of $\mul$, defined as $$ \begin{array}{rl} \mul_b({\calF})\!\!\! & =\min\left\{\left|{\mathcal{H}}\right|\colon {\mathcal{H}}\in \op{UB}\ \&\ \neg\left(\exists g\in{\mathcal{B}},\, g\neq 0\right) \left(\forall h\in{\mathcal{H}}\right)\left(g\cdot h\in{\calF}\right)\right\}. \end{array} $$ However this function has not been studied so far. One might also consider the study of the operator $\add$ (and $\mul$) for the functions from $\real^n$ into $\real$ with $n>1$. This has indeed been done by Ciesielski and Wojciechowski in~\cite{CW}. The study concerned only the classes $\ext(\R^n)$, $\ac(\R^n)$, $\conn(\R^n)$, $\D(\R^n)$, and $\pc(\R^n)$ since other classes from Chart~2 do not have natural generalizations into functions of more than one variable. First, one should recall that for $n>1$ Chart~1 is not valid any more. The new inclusions (for $n>1$) are as follows: $$ \ext(\R^n)\subset\pc(\R^n)=\conn(\R^n)\subset\D(\R^n)\cap\ac(\R^n), $$ $$ \D(\R^n)\not\subset\ac(\R^n),\ \ \ \ac(\R^n)\not\subset\D(\R^n),\ \ \ \D(\R^n)\cap\ac(\R^n)\not\subset\conn(\R^n). $$ (The inclusion ``$\pc(\R^n)\subset\conn(\R^n)$'' was proved by Hamilton~\cite{Hamilton} and by Stallings~\cite{Stal}, and the inclusion ``$\conn(\R^n)\subset\pc(\R^n)$'' by Hagan~\cite{Hagan}. The proof of the inclusion ``$\conn(\R^n)\subset\ac(\R^n)$'' is presented in~\cite{Stal}. The examples showing that $\D(\R^n)\not\subset\ac(\R^n)$ and $\ac(\R^n)\not\subset\D(\R^n)$ can be found in \cite[Examples 1.1.9 and 1.1.10]{N1hab} or \cite[Examples 1.7 and 1.6]{N1}, while a simple Baire class 1 function in $\D(\R^n)\cap\ac(\R^n)\setminus\conn(\R^n)$ was described in \cite[Example~1]{RGR}.) We do not know whether the inclusion $\ext(\R^n)\subset\pc(\R^n)$ is proper. The problem with studying the value of the operator $\add$ for all these classes (except for $\ac(\R^n)$) is that there exists a function $f\colon\real^n\to\R$ which is not a sum of $n$ Darboux functions, implying that \[ \add(\ext(\R^n))=\add(\pc(\R^n))=\add(\conn(\R^n))=\add(\D(\R^n))=2. \] However, every function $f\colon\R^n\to\R$ is a sum of $n+1$ extendable functions. To express these results nicely, define for $\F\subset\R^{\R^n}$ the {\em repeatability} ${\mathcal{R}}({\cal F})$ of ${\cal F}$ as the smallest integer $k$ such that any function $f\colon \R^n\rightarrow\R$ can be expressed as a sum of $k$ functions from ${\cal F}$. (We put ${\mathcal{R}}({\cal F})=\infty$ if such a number does not exist.) In this language the results of Ciesielski and Wojciechowski can be stated as follows. \medskip \begin{theorem}[Ciesielski, Wojciechowski~\cite{CW}]%4.27 \label{th:CWoj} \[ {\mathcal{R}}(\ext(\R^n))={\mathcal{R}}(\conn(\R^n))={\mathcal{R}}(\pc(\R^n))=n+1. \] \end{theorem} \medskip \noindent Clearly Theorem~\ref{th:CWoj} implies that ${\mathcal{R}}(\D(\R^n))\leq n+1$. The problem (stated in~\cite{CW}) whether this equation can be replaced by the equality has been recently solved by F.~Jordan. \medskip \begin{theorem}[F.~Jordan~\cite{FjordanPr}] %4.28 \label{th:JorDarb} For every $n$ there exists a Baire~1 class function $f\colon\R^n\to\R$ which is not a sum of $n$ Darboux functions. In particular, \[ {\mathcal{R}}(\D(\R^n))=n+1. \] \end{theorem} \medskip The value of ${\mathcal{R}}(\ac(\R^n))$ is clearly equal to $2$, since Natkaniec~\cite{N1} proved that $\add(\ac(\R^n))>\co$. This fact has been recently improved by F.~Jordan, who proved \medskip \begin{theorem} [F.~Jordan~\cite{FjordanPr}]%4.29 \label{th:JorACRn} For every $n\geq 1$ \[ \add(\ac(\R^n))=\add(\ac)={\frak e}_\continuum. \] \end{theorem} \medskip \noindent In~\cite{FjordanPr} F.~Jordan considers also the following version of additivity function for the classes $\F$ of functions from $\R^n$ into $\R$ \begin{equation*} \genadd_{n,k}(\F)= \min\left(\left\{|G|\colon G\subseteq \real^{\R^n} \&\ \Psi_{n,k}(G) \text{ holds}\right\} \cup\left\{\left(2^\continuum\right)^+\right\}\right) \end{equation*} where $k\F=\{f_0+\cdots+f_{k-1}\colon f_i\in\F\}$ and $\Psi_{n,k}(G)$ denotes the statement \begin{equation*} (\forall f\in (n-k)\F) (\exists g\in G)(g-f\notin (k+1)\F). \end{equation*} \newpage This function makes a good generalization of both functions $\add$ and ${\mathcal{R}}$ since \[ {\mathcal{R}}(\F) = n+1\ \text{ if and only if } \ 1<\genadd_{n,k}(\F)<\left(2^\continuum\right)^+ \text{ for some } k