\documentclass[12pt]{article}%\usepackage{amstex} \usepackage{amsmath}\usepackage{amssymb} %\newcommand{\restriction}{{\hbox{$\,|\grave{}\,$}}} \def\continuum{{\mathfrak c}} \def\co{\continuum} % characteristic function \newcommand{\charf}[1]{\mbox{\raise.48ex\hbox{$\chi$}$_{#1}$}} \def\proof{\noindent{\sc Proof. }} %\def\qed{\hfill$\Box$} \def\qed{\hfill\vrule height6pt width6pt depth1pt \medskip} \newcommand{\Qed}{\qed} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{fact}[theorem]{Fact} \newtheorem{problem}{Problem}[section] \newtheorem{example}[theorem]{Example} \newtheorem{definition}{Definition} \newtheorem{conjecture}{Conjecture} \newcommand{\thm}[2]{\begin{theorem}\label{#1}#2\end{theorem}} \newcommand{\cor}[2]{\begin{corollary}\label{#1}#2\end{corollary}} \newcommand{\prop}[2]{\begin{proposition}\label{#1}#2\end{proposition}} \newcommand{\lem}[2]{\begin{lemma}\label{#1}#2\end{lemma}} \newcommand{\fac}[2]{\begin{fact}\label{#1}#2\end{fact}} \newcommand{\pr}[2]{\begin{problem}\label{#1}#2\end{problem}} \newcommand{\ex}[2]{\begin{example}\label{#1}#2\end{example}} \newcommand{\defi}[2]{\begin{definition}\label{#1}#2\end{definition}} \newcommand{\conj}[2]{\begin{conjecture}\label{#1}#2\end{conjecture}} \newcommand{\bd}{{\rm bd}} %\newcommand{\mathR}{\/\mbox{\rm I\kern -0.18em{R}}} \newcommand{\mathR}{{\mathbb R}} \newcommand{\real}{\mathR} \newcommand{\rational}{{\mathbb Q}} %\newcommand{\mathI}{\/\mbox{\rm I\kern -0.18em{I}}} \newcommand{\mathI}{{\mathbb I}} %\newcommand{\mathN}{\/\mbox{\rm I\kern -0.18em{N}}} \def\lin{{\operatorname{LIN}_\rational}} \def\proj{{\operatorname{pr}}} \def\supp{{\operatorname{supp}}} \newcommand{\F}{{\cal F}} \newcommand{\e}{\varepsilon} \newcommand{\J}{{\cal J}} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \def\P{{\cal P}} \newcommand{\A}{{\cal A}} \newcommand{\B}{{\cal B}} \newcommand{\C}{{\cal C}} \newcommand{\K}{{\cal K}} \newcommand{\X}{{\cal X}} \newcommand{\Ma}{{\cal M}_{a}} \newcommand{\Mm}{{\cal M}_{m}} \newcommand{\Ml}{{\cal M}_{l}} \newcommand{\Const}{{\rm Const}} \newcommand{\G}{{\cal G}} \newcommand{\I}{{\cal I}} \newcommand{\CC}{{\rm C}} \newcommand{\PC}{{\rm PC}} \newcommand{\D}{{\rm D}} \newcommand{\M}{{\rm M}} \newcommand{\AC}{{\rm AC}} \newcommand{\Conn}{{\rm Conn}} \newcommand{\Add}{{\rm Add}} \newcommand{\darb}{{\rm D}} \newcommand{\EE}{{\cal E}} \newcommand{\Ext}{{\rm Ext}} \newcommand{\PR}{{\rm PR}} \newcommand{\CIVP}{{\rm CIVP}} \newcommand{\SCIVP}{{\rm SCIVP}} \newcommand{\WCIVP}{{\rm WCIVP}} \newcommand{\DIVP}{{\rm DIVP}} \newcommand{\bor}{{{\cal B}{\rm or}}} %\newcommand{\E}{{\rm E}} \newcommand{\cl}{{\rm cl}} \newcommand{\dist}{{\rm dist}} \newcommand{\diam}{{\rm diam}} \title{Darboux-like functions within the classes of Baire one, Baire two, and additive functions} \author{ Krzysztof Ciesielski% \thanks{ 1991 {\it Mathematics Subject Classification}. Primary 26A15, 54C30; secondary 26A21, 54C08 \endgraf %\newline \hspace*{4pt} {\it Key words and phrases}. Extendable, almost continuous, connectivity, additive functions. \endgraf %\newline \hspace*{4pt} This work was partially supported by NSF Cooperative Research Grant INT-9600548 with its Polish part being financed by Polish Academy of Science PAN. The results were obtained during a visit of the second author at West Virginia University. }\\ {\footnotesize Department of Mathematics}\\ {\footnotesize West Virginia University}\\ {\footnotesize Morgantown, WV 26506-6310}\\ {\footnotesize USA}\\ {\footnotesize KCies@wvnvms.wvnet.edu} %Department of Mathematics, West %Virginia University, Morgantown, WV 26506-6310, USA (kcies@wvnvms.wvnet.edu) \and Jan Jastrz{\c{e}}bski$^*$\\ {\footnotesize Department of Mathematics}\\ {\footnotesize Gda{\'n}sk University}\\ {\footnotesize Wita Stwosza 57}\\ {\footnotesize 80-952 Gda{\'n}sk, Poland}\\ {\footnotesize jjas@ksinet.univ.gda.pl}} \date{} \begin{document} \maketitle \begin{abstract} In the paper we present an exhaustive discussion of the relations between Darboux-like functions within the classes of Baire one, Baire two, Borel, and additive functions from $\real^n$ into $\real$. In particular we construct an additive extendable discontinuous function $f\colon\real\to\real$, answering a question of Gibson and Natkaniec \cite[p.~499]{GN}, and show that there is no similar function from $\real^2$ into $\real$. We also describe a Baire class two almost continuous function $f\colon\real\to\real$ which is not extendable. This gives a negative answer to a problem of Brown, Humke, and Laczkovich \cite[prob.~1]{BHL}. (See also \cite[prob.~3.21]{GN}.) %Finally, assuming the continuum hypothesis, we will construct %an additive connectivity function $f\colon\real\to\real$ %which is not almost continuous. This gives a partial answer to another %question of Gibson and Natkaniec \cite[prob.~5.5]{GN}. \end{abstract} \section{Introduction} The study of different generalizations of continuity of functions from $\real^n$ into $\real$ has a long history. In this paper we will be interested in the functions with some of these generalized continuities that are known under the common name of Darboux-like functions. The readers unfamiliar with their definitions can find them in the next section. The basic relations between these classes, for the functions from $\real$ to $\real$, are given in the following chart, in which arrows $\longrightarrow$ denote strict inclusions. Moreover, all other possible ``natural intersection'' inclusions (in a form of $\AC\cap\CIVP\subseteq\Conn\cap\CIVP$) obtained from different classes of this chart remain strict. \hspace{1.5pc} \begin{picture}(0,90) \put(20,55){\makebox(0,0){$\CC$}} \put(35,55){\vector(1,0){20}} \put(75,55){\makebox(0,0){$\Ext$}} \put(88,60){\vector(2,1){18}} \put(120,70){\makebox(0,0){${\AC}$}} \put(132,70){\vector(1,0){30}} \put(180,70){\makebox(0,0){$\Conn$}} \put(195,70){\vector(1,0){20}} \put(224,70){\makebox(0,0){${\D}$}} \put(233,65){\vector(2,-1){18}} \put(270,55){\makebox(0,0){${\PC}$}} \put(127,40){\makebox(0,0){${\SCIVP}$}} \put(180,40){\makebox(0,0){${\CIVP}$}} \put(225,40){\makebox(0,0){${\PR}$}} \put(145,40){\vector(1,0){20}} \put(195,40){\vector(1,0){20}} \put(88,52){\vector(2,-1){18}} \put(233,43){\vector(2,1){18}} \put(195,38){\vector(1,-1){20}} \put(235,15){\makebox(0,0){${\WCIVP}$}} \end{picture} \begin{center} Chart~1 \end{center} The inclusions $\CC\subset\Ext$, $\Conn\subset\darb\subset\PC$, $\SCIVP\subset\CIVP\subset\WCIVP$, and $\PR\subset\PC$ are obvious from the definitions. The inclusions $\Ext\subset\AC\subset\Conn$ were proved by Stallings~\cite{JS}. The inclusion $\CIVP\subset\PR$ was stated without the proof in \cite{GRou}. The proof can be found in \cite[thm.~3.8]{GN}. The inclusion $\Ext\subset\SCIVP$ was proved by Rosen, Gibson, and Roush in~\cite{RGR}. An excellent discussion of this chart can be found in a recent survey by Gibson and Natkaniec~\cite[sec.~3]{GN}. The examples concerning the properness of all intersection inclusions can be also found, in stronger versions, in Theorems~\ref{thA}, \ref{thB}, and \ref{thE} stated below. Also, function $F$ from Corollary~\ref{cor3} is the first simple ZFC example of almost continuous SCIVP function which is not extendable. For the functions from $\real^n$ into $\real$ with $n>1$ the classes from the lower part of Chart~1 are not defined. The relations between the classes in the upper part of the chart change to the following. \begin{picture}(0,90) \put(0,55){\makebox(0,0){$\CC(\real^n)$}} \put(19,55){\vector(1,0){15}} \put(120,55){\makebox(0,0){$\Ext(\real^n)=\Conn(\real^n)=\PC(\real^n)$}} \put(205,55){\vector(1,0){15}} \put(265,55){\makebox(0,0){$\AC(\real^n)\cap\D(\real^n)$}} \put(310,60){\vector(2,1){18}} \put(352,70){\makebox(0,0){${\AC(\real^n)}$}} \put(350,40){\makebox(0,0){${\D(\real^n)}$}} \put(310,52){\vector(2,-1){18}} \end{picture} \vspace{-2pc}\begin{center} Chart~2 \end{center} The inclusions $\CC(\real^n)\subset\Ext(\real^n)\subset\Conn(\real^n)$ are obvious from the definitions. The inclusion $\Conn(\real^n)\subset\Ext(\real^n)$ was recently proved by Ciesielski, Natkaniec, and Wojciechowski~\cite{CNW}. The containment $\Conn(\real^n)\subset\PC(\real^n)$ was proved by Hamilton~\cite{OH} and Stallings~\cite{JS}, and the inclusion $\PC(\real^n)\subset\Conn(\real^n)$ by Hagan~\cite{H}. (See also Whyburn~\cite{Why} and \cite[thm,~8.1]{GN}.) The relation %inclusion $\Conn(\real^n)\subset\AC(\real^n)\cap\darb(\real^n)$ was proved by Stallings~\cite{JS}. The examples concerning the properness of the inclusions can be found, in the Baire class one, in Theorem~\ref{thC}. The main goal of this paper is to discuss these two charts when we restrict the function in all these classes to the following four classes of functions: Baire one $B_1$, Baire two $B_2$, Borel $\bor$, and additive functions $\Add$. Notice that an intersection of any two of these classes is trivial, since $B_1\subset B_2\subset\bor$ and $\Add\cap\bor\subset\CC$. \thm{thA}{For the Baire one functions $B_1$ from $\real$ to $\real$ the following holds. \[ \CC\subsetneq\Ext=\AC=\Conn=\darb=\PC=\SCIVP=\CIVP=\PR\subsetneq\WCIVP. \] } \proof The proof of the equation $\Ext\cap B_1=\PC\cap B_1$ can be found in~\cite{BHL}. This equation and Chart~1 imply all the other equations. The properness of the inclusions is justified as follows. \begin{description} \item{$B_1\cap\darb\setminus\CC\neq\emptyset$ --} It is witnessed by the function $f_0\colon\real\to\real$ defined by $f_0(x)=\sin(1/x)$ for $x\neq 0$ and $f_0(0)=0$. (See \cite[example~1.1]{N1}.) \item{$B_1\cap\WCIVP\setminus\PC\neq\emptyset$ --} It is witnessed by the function $g\colon\real\to\real$ defined by $g(x)=x^2$ for $x\neq 0$ and $g(0)=-1$.\qed \end{description} \thm{thB}{The classes from Chart~1 restricted to either Baire two functions $B_2$ or Borel functions from $\real$ to $\real$ leads to the following chart. Moreover, all possible ``natural intersection'' inclusions obtained from different classes of this chart remain strict. \begin{picture}(0,90) \put(130,55){\makebox(0,0){$\CC%\longrightarrow \longrightarrow\Ext\longrightarrow\AC\longrightarrow \Conn\longrightarrow\darb\longrightarrow\SCIVP=\CIVP$}} \put(280,60){\vector(2,1){18}} \put(332,70){\makebox(0,0){$\PR\longrightarrow\PC$}} \put(328,40){\makebox(0,0){${\WCIVP}$}} \put(280,52){\vector(2,-1){18}} \end{picture}\vspace{-2pc} } \proof To see that the inclusion $\darb\subseteq\SCIVP$ holds in the class of Borel functions let $x1$, Chart~2 remains unchanged.} \proof The properness of all the inclusions, as well as of their other possible combinations, is justified by the following facts. \begin{description} \item{$B_1(\real^n)\cap\Conn(\real^n)\setminus\CC(\real^n)\neq\emptyset$ --} It is witnessed by a function $f\colon\real^2\to\real$ given by $f(x,y)=\sin(1/(x^2+y^2))$ for $\la x,y\ra\neq \la 0,0\ra$ and $f(0,0)=0$. It is in $\PC(\real^2)$ straight from the definition. \item{$B_1(\real^n)\cap\AC(\real^n)\cap \darb(\real^n)\setminus\Conn(\real^n)\neq\emptyset$ --} Rosen, Gibson, Roush~\cite[example~1]{RGR} proved that a function $f\colon[-1,1]\times[0,1]\to[-1,1]$ given by $f(x,y)=\sin(1/y)$ for $y>0$ and $f(x,0)=x$ is Baire one, almost continuous, Darboux, but not connectivity. It is easy to extend it to a finite support function on $\real^2$ with the same properties. \item{$B_1(\real^n)\cap\darb(\real^n)\setminus\AC(\real^n)\neq\emptyset$ --} It is justified by the function $F(x,y)=f_0(x)$, where $f_0$ is a function from Theorem~\ref{thA}. (See Natkaniec~\cite[example~1.7]{N1} or \cite[example~1.1.9]{N1hab}.) \item{$B_1(\real^n)\cap\AC(\real^n)\setminus\darb(\real^n)\neq\emptyset$ --} Let $f(x)=\sin(1/x)$ for $x\neq 0$ and $f(0)=1$, and let $F\colon[-1,1]^2\to[-1,1]$ be given by the formula $F(x,y)=y\ f(x)$. It was proved by Natkaniec \cite[example~1.1.10]{N1hab} that $F$ is Baire one, almost continuous, and not Darboux. It is easy to extend $F$ to a function $\bar F\colon\real^2\to[0,1]$ with compact support while preserving these properties. \qed \end{description} The study of classes of additive functions from Charts~1 and~2 were initiated by D.~Banaszewski~\cite{Ban}. (See also \cite[sec.~5]{GN}.) In this direction we have the following results. \thm{thD}{For the additive functions $\Add(\real^n)$ from $\real^n$ to $\real$, $n>1$, Chart~2 changes as follows: \hspace{6pt}\begin{picture}(0,90) \put(140,55){\makebox(0,0){$\CC(\real^n)=\Ext(\real^n)=\Conn(\real^n)=\PC(\real^n) =\AC(\real^n)\cap\D(\real^n)$}} \put(300,60){\vector(2,1){18}} \put(345,70){\makebox(0,0){${\AC(\real^n)}$}} \put(342,40){\makebox(0,0){${\D(\real^n)}$}} \put(300,52){\vector(2,-1){18}} \end{picture} \vspace{-2pc} } \proof The inclusion $\Add(\real^n)\cap\AC(\real^n)\cap\D(\real^n)\subset\CC(\real^n)$ is proved in Theorem~\ref{thm2}. The properness of the inclusions is justified by the following facts. \begin{description} \item{$\Add(\real^n)\cap\AC(\real^n)\setminus\darb(\real^n)\neq\emptyset$ --} See Example~\ref{ACnotD}. \item{$\Add(\real^n)\cap\darb(\real^n)\setminus\AC(\real^n)\neq\emptyset$ --} See Example~\ref{DnotAC}.\qed \end{description} \thm{thE}{For the additive functions $\Add$ from $\real$ to $\real$ we have the equation $\PR=\WCIVP$. The other inclusions of Chart~1 remain unchanged, except possibly for the inclusion $\AC\subset\Conn$. Thus, we have \hspace{1.5pc} \begin{picture}(0,90) \put(20,55){\makebox(0,0){$\CC$}} \put(35,55){\vector(1,0){20}} \put(75,55){\makebox(0,0){$\Ext$}} \put(88,60){\vector(2,1){18}} \put(120,70){\makebox(0,0){${\AC}$}} \put(132,70){\vector(1,0){30}} \put(180,70){\makebox(0,0){$\Conn$}} \put(195,70){\vector(1,0){20}} \put(224,70){\makebox(0,0){${\D}$}} \put(233,65){\vector(2,-1){18}} \put(270,55){\makebox(0,0){${\PC}$}} \put(127,40){\makebox(0,0){${\SCIVP}$}} \put(180,40){\makebox(0,0){${\CIVP}$}} \put(225,40){\makebox(0,0){${\PR}$}} \put(145,40){\vector(1,0){20}} \put(195,40){\vector(1,0){20}} \put(88,52){\vector(2,-1){18}} \put(233,43){\vector(2,1){18}} \end{picture} \vspace{-2pc} %\begin{center} Chart~4 \end{center} \noindent Moreover all possible ``natural intersection'' inclusions obtained from the different classes of this chart and not involving $\AC\subset\Conn$ remain strict. The inclusion $\Add\cap\CIVP\cap\AC\subset\Conn$ is strict if union of less than continuum many meager subsets of $\real$ is meager in $\real$. } \proof The properness of all the inclusions is justified by the following facts. \begin{description} \item{$\Add\cap\Ext\setminus\CC\neq\emptyset$ --} See Corollary~\ref{cor1}. \item{$\Add\cap\SCIVP\cap\AC\setminus\Ext\neq\emptyset$ --} See Ciesielski, Ros{\l}anowski~\cite{CRo}. Compare also Ciesielski~\cite[thm.~3.1]{SomeDarbF}. \item{$\Add\cap\SCIVP\cap\darb\setminus\Conn\neq\emptyset$ --} See Example~\ref{SCIVPandDnotCon}. (An example of a function from $\Add\cap\darb\setminus\Conn$ was earlier given in~\cite{Ban}.) \item{$\Add\cap\PC\setminus\darb\neq\emptyset$ --} See Example~\ref{SCIVPandPCnotD}. (An example of a function from $\Add\cap\CIVP\setminus\darb$ was earlier given in~\cite{Ban}.) \item{$\Add\cap\AC\cap\CIVP\setminus\SCIVP\neq\emptyset$ --} See Ciesielski~\cite[thm.~4.1]{SomeDarbF}. \item{$\Add\cap\AC\cap\PR\setminus\CIVP\neq\emptyset$ --} See Example~\ref{ACandPRnotCIVP}. \item{$\Add\cap\AC\setminus\PR\neq\emptyset$ --} See D.~Banaszewski~\cite{Ban}. \item{$\Add\cap\CIVP\cap\Conn\setminus\AC$ --} Such a function, under the assumption that union of less than continuum many meager subsets of $\real$ is meager, has been constructed by Ciesielski and Ros{\l}anowski~\cite{CRo}. (Example for $\Add\cap\Conn\setminus\AC$ requires only that union of less than continuum many meager subsets of $\real$ does not cover $\real$.) \qed \end{description} The following questions, which are variants of D.~Banaszewski's question \cite[Question~5.5]{GN}, remain open. \pr{prob1}{ \begin{description} \item[(1)] Does there exist a ZFC example of an additive connectivity function $f\colon\real\to\real$ which is not almost continuous? \item[(2)] Does there exist an additive connectivity function $f\colon\real\to\real$ with $\SCIVP$ property which is not almost continuous? \end{description} } \section{Definitions and notation} Our terminology is standard and follows \cite{Ci:book}. We consider only real-valued functions of one or more real variables. No distinction is made between a function and its graph. A restriction of a function $f\colon X\to Y$ to a set $A\subset X$ is denoted by $f\restriction A$. Symbol $\charf{A}$ will be used for a characteristic function of a subset $A$ of a fixed space $X$. By $\real$ and $\rational$ we denote the set of all real and rational numbers, respectively. We will consider $\real^n$ as linear spaces over $\rational$. In particular, for $X\subset\real^n$ we will use the symbol $\lin(X)$ to denote the smallest linear subspace of $\real^n$ over $\rational$ that contains $X$. Recall also that if $D\subset\real^n$ is linearly independent over $\rational$ and $f\colon D\to\real$ then $F=\lin(f)\subset\real^{n+1}$ is an additive function (see definition below) from $\lin(D)$ into $\real$ which extends $f$. Any linear basis of $\real$ over $\rational$ will be referred as a {\em Hamel basis}. The ordinal numbers will be identified with the sets of all their predecessors and cardinals with the initial ordinals. In particular $2=\{0,1\}$ and the first infinite ordinal $\omega$ number is equal to the set of all natural numbers $\{0,1,2,\ldots\}$. The family of all functions from a set $X$ into $Y$ is denoted by $Y^X$. In particular $2^n$ will stand for the set of all sequences $s\colon \{0,1,2,\ldots,n-1\}\to\{0,1\}$, while $2^{<\omega}=\bigcup_{n<\omega}2^n$ is the set of all finite sequences into $2$. The symbol $|X|$ stands for the cardinality of a set $X$. The cardinality of $\real$ is denoted by $\co$ and referred as {\em continuum}. A set $S\subset\real$ is said to be {\em $\co$-dense\/} if $|S\cap(a,b)|=\co$ for every $a1$ then $f$ is connectivity. This follows from the following well known fact. \fac{fact1}{If $S\subset\real^2$ disconnects $\real^2$ then it contains a nontrivial compact connected subset.} A function $f\colon\real\to\real$ has: \begin{itemize} \item {\it Cantor intermediate value property} if for every $x,y\in\mathR$ and for each perfect set $K$ between $f(x)$ and $f(y)$ there is a perfect set $C$ between $x$ and $y$ such that $f[C]\subset K$; \item {\it strong Cantor intermediate value property\/} if for every $x,y\in\mathR$ and for each perfect set $K$ between $f(x)$ and $f(y)$ there is a perfect set $C$ between $x$ and $y$ such that $f[C]\subset K$ and $f\restriction C$ is continuous; \item {\it weak Cantor intermediate value property} if for every $x,y\in\real$ with $f(x)0$ less than the diameters of $I$ and $B_s$ and such that \begin{description} \item{($*$)} for every $J\in\J$ with $J\subset I$ if the distance from $J$ to $p_s$ is less than $\e$ then $J\times\{0\}\subset U_s$. %$z\in I$ with $|z-p_s|<\e$ and for every $p\in[0,1]$ such that %either $p=z$ or $p,z\in\cl(J)$ for some $J\in\J$ we have $\la p,0\ra\in U_s$. \end{description} Let $W_s\subset[0,1]^2$ be an open neighborhood of $\la p_s,0\ra$ with diameter less than $\e$ and such that $F[\bd(W_s)]\subset[1-2^{-n},1]$. It exists since $F$ is peripherally continuous at $\la p_s,0\ra$ and $F(p_s,0)=f(p_s)=1$. Without loss of generality we can assume that $\bd(W_s)$ is connected. (Replacing $W_s$ by its component, if necessary, we can assume that $W_s$ is connected. Then we can increase $W_s$ to $[0,1]^2\setminus V_s$, where $V_s$ is an ``unbounded'' component of $[0,1]^2\setminus W_s$, that is the one which contains the boundary of $[0,1]^2$. This decreases $\bd(W_s)$ and makes it connected.) Note that the choice of $\e$ guarantees that $\bd(W_s)\cap B_s\neq\emptyset\neq\bd(W_s)\cap (I\times\{0\})$ since $B_s$ and $I\times\{0\}$ are connected and $\bd(W_s)$ disconnects $[0,1]^2$. Let $z\in I$ be such that $\la z,0\ra\in \bd(W_s)\cap (I\times\{0\})$. Since $F(z,0)\in F[\bd(W_s)]\subset[1-2^{-n},1]\subset(0,1]$ there exists a $J\in\J$ such that $z\in\cl(J)$. Let $p_t$ be the endpoint of $\cl(J)$ for which $f(p_t)=1$. By ($*$) we have $\la p_t,0\ra\in U_s$. The set $B_t$ is defined as a union of $\bd(W_s)$ and a closed segment joining $\la z,0\ra$ and $\la p_t,0\ra$. The open neighborhood $U_t$ of $p_t$ is chosen such that its diameter is at most $2^{-(n+1)}$ and that $p_s\notin\cl(U_t)\subset U_s$. It is easy that conditions (i), (ii), and (iii) are satisfied. To choose $B_{t'}$, $p_{t'}$, and $U_{t'}$ we replace $U_s$ with $U^\prime_s=U_s\setminus \cl(U_t)$ and repeat the process described above. This finishes the inductive construction. Now, to finish the argument take an arbitrary $s\in 2^\omega$ and note that by (i) and (iii) the limit $\lim_{n\to\infty}\la p_{s\restriction n},0\ra$ exists and is equal to a point $p_s$ which belongs to $C\times\{0\}$. Also, by (ii), the set $B_s=\bigcup_{00$ is less then the diameter of $B_s$ and let $U$ be an open neighborhood of $p_s$ of diameter less than $\e$ and such that $F[\bd(U)]\subset[0,1/2)$. Then for a point $w\in B_s\cap\bd(U)$ we have $F(w)\in[0,1/2)\cap[1/2,1]$, a contradiction. This finishes the proof that $f$ is not extendable. Next we will show that $f$ is almost continuous. Let $G$ be an open set contained in $[0,1]^2$ containing the graph of $f$. For every $x\in[0,1]$ there exists an interval $(a_x,b_x)$ such that \begin{itemize} \item $x\in(a_x, b_x)$, \item $f(a_x)=f(b_x)=0$, and \item there is a continuous function $g_x \colon [0,1] \to \mathR$ with $g_x \restriction{[a_x,b_x]} \subset G$ and such that $g_x(t)=0$ for $t \notin (a_x, b_x)$. \end{itemize} Indeed, if $f(x)=0$ then it is easy to find $(a_x, b_x)$ for which $g_x\equiv 0$ works. If $f(x)\neq 0$ then $x\in J$ for some $J\in\J$, say $J=(A,B)$. Assume that $f$ is increasing on $J$, the other case is similar. Then $f(B)=1$. Find an interval $J'=(C,D)\in\J$ on which function $f$ is decreasing and such that $[B,C]\times\{1\}\subset G$. Put $a_x=A$, $b_x=D$, define $g_x(B)=g_x(C)=1$, $g_x(0)=g_x(a_x)=g_x(b_x)=g_x(1)=0$, and extend $g_x$ in a linear way on each interval with these endpoints. Thus $g_x$ has a hat shape. Now choose a finite subcover $\{(a_{x_i},b_{x_i})\colon i0$ there exists an $n<\omega$ such that $x\in I_n$, $g(x)\in J_n$, and $\max\{\diam(I_n),\diam(J_n)\}<\e$, \item{(d)} for every $n0$ then for every $\la t,I,J\ra\in F_n$ there exists $\la t_0,I_0,J_0\ra\in F_{n-1}$ such that $I\subset I_0$. \end{description} The induction can be started with $F_0=\{\la t_0,(0,1),(0,1)\ra\}$, and can be easily carried through since $C$ is compact zero dimensional and the sets $f^{-1}(1)$ and $T\cap f^{-1}(0)$ are dense in $C$. Now, let $\{\la t_n,I_n,J_n\ra\colon n<\omega\}$ be an enumeration of $\bigcup_{n<\omega}F_n$. We claim that $\la \la I_n,J_n\ra\colon n<\omega\ra$ is a PI family for $f_0=f+\charf{B}$ for an appropriately chosen set $B$. Clearly (3) and (5) imply that condition (d) is satisfied in void. Condition (a) can be deduced from the density of $T$ in $C$ and (1--3). (b) for $g=f_0$ is implied by (2) and the fact that $f_0(x)=f(x)$ for $x\in[0,1]\setminus C$. Similarly (c) for the points $x\in T$ is implied by (4) and $f_0\restriction T=f\restriction T$. To finish the proof it is enough to show that (c) holds for points $x\in C_0$ for an appropriate choice of $B$. But every $x\in C_0$ the set $S_x=\{k<\omega\colon x\in I_k\}$ is infinite. Let $B$ be the set of all those points $x\in C_0$ for which the set $\{k\in S_x\colon f(t_k)=0\}$ is finite. Then for every $x\in C_0$ the set $\{k\in S_x\colon f_0(x)\in J_k\}$ is infinite proving (b). The fact that $B$ is a meager $F_\sigma$ subset of $C_0$ is left as an exercise. \qed \ex{B2SCIVPnotD}{There exists an $f\in B_2\cap\SCIVP\setminus\darb$. } \proof Define $g$ from $[-1,1]$ onto $(0,1]$ by $g(x)=(x^2-1)\sin^2(1/x)+1$ for $x\neq 0$ and $f(0)=1$. Note that $g(-1)=g(1)=1$ and that $g$ is $\SCIVP$. For each component $J$ of $[0,1]\setminus C$ of length $1/3^{-n}$ define $f\restriction\cl(J)$ as $(-1)^n g\circ h_J$, where $h_J$ is an increasing linear function with $h_J[\cl(J)]=[-1,1]$. For all other points $x$ we put $f(x)=1$. Note that $f[\real]=[-1,0)\cup(0,1]$ and that preimage of every open set is a union of a $G_\delta$ and an $F_\sigma$ set. So $f$ is Baire two and not Darboux. To see that it is $\SCIVP$ take $x1$. Its proof will be based on the following two propositions, the first of which was proved by Lipi\'{n}ski~\cite{Lip}. (See also Maliszewski and Natkaniec~\cite{MN}.) This fact was noticed independently by the authors of this paper and our proof is enclosed below. \prop{propA}{If $f\colon\real\to\real$ is discontinuous then $F\colon\real^2\to\real$ given by $F(x,y)=f(x)$ is not almost continuous. } \proof Assume that $f\colon\real\to\real$ is discontinuous at some point $x$. Taking a translation of a graph of $f$, if necessary, we can assume that $x=0$ and $f(0)=0$. So, there exists a sequence $\{x_n\}_{n<\omega}$ converging to $0$ such that $\lim_{n\to\infty}f(x_n)=L_0\neq 0$. Multiplying $f$ by $3/L_0$, if necessary, we can assume that $L_0=3$. We will also assume that $f(x_n)>2$ for all $n<\omega$. Consider the closed set \[ A=\{\la x,y,z\ra\colon x=x_n\mbox{ for some } n<\omega, y\in\real, \mbox{ and }z\leq 2\}\cup(\{0\}\times\real^2) \] and let $G=\real^3\setminus A$. Then $G$ is an open set containing all the graph of $F$ except of a line $L=\{0\}\times\real\times\{0\}$. Let $H=\{\la x,y,z\ra\in\real^3\colon x^2+z^20$ is such that $e^{-(|x_n|+b)^2}=1$. Moreover, $\la x_n,0,g(x_n,0)\ra\in \{x_n\}\times H_0$. But clearly there is no continuous function on $\real$ whose graph is contained in $G_0\cup H_0$ and intersects $H_0$. This contradiction finishes the proof. \qed Notice that Proposition~\ref{propA} stay in contrast with the following fact. \fac{factContr}{ {\rm (Natkaniec~\cite[cor.~4.2(1)]{N1})} If $f\colon\real\to\real$ is almost continuous and $Y$ is a compact topological space then $F\colon\real\times Y\to\real$ given by $F(x,y)=f(x)$ is almost continuous.} \prop{propB}{If $n>1$ and $F\in\Add(\real^n)\cap\darb(\real^n)$ then $F^{-1}(0)$ contains a straight line. } \proof If $F$ is constantly equal to $0$ then there is nothing to prove. So, assume that this is not the case. Then $f[\real^2]=\real$. In particular $V=F^{-1}(0)$ disconnects $\real^2$. (Otherwise its complement $\real^2\setminus V$ would be connected, while $F[\real^2\setminus V]=\real\setminus\{0\}$ would not, contradicting Darboux property.) Thus, by Fact~\ref{fact1}, $V$ contains a nontrivial compact connected subset $K$. Pick $a,b\in K$ with $\diam(\{a,b\})=\diam(K)$. Since every rotation $r$ is a linear homeomorphism, replacing $F$ with $F\circ r$ for an appropriate $r$ if necessary, we can assume that $a$ and $b$ are on the same vertical line. If $\proj(K)$ is a singleton, then $K$ contains a straight line segment connecting $a$ with $b$. This, and the fact that $V$ is linear over $\rational$, easily imply that $V$ contains a straight line. So, assume that $\proj(K)=[x_0,x_1]$ for some $x_00$ such that some open disk of radius $\e$ is contained in $B$. Then, if $B(\e)=\{v\in\real^2\colon\mbox{ length of $v$ is less than }\e\}$ then for every $v\in B(\e)$ we have $\bd(W)\cap(v+\bd(W))\neq\emptyset$. So, $v\in\bd(W)-\bd(W)\subset V$. Therefore, $B(\e)\subset V$, and so $V=\real^2$ contradicting our assumption that $F$ is non-zero. \qed \thm{thm2}{$\Add(\real^n)\cap\AC(\real^n)\cap\darb(\real^n)\subset\CC(\real^n)$ for $n>1$. } \proof Let $F\in \Add(\real^n)\cap\AC(\real^n)\cap\darb(\real^n)$. We will show that it is continuous. Since it is enough to prove continuity of any restriction of $F$ to a plane containing the origin, we can assume that $n=2$. Now, by Proposition \ref{propB}, $f^{-1}(0)$ contains a line $L_0$. Since $f^{-1}(0)$ is close under addition, it contains also a parallel line $L$ which contains the origin. Since all the classes under consideration are closed under inner composition with a rotation, we can assume that $L$ is a vertical line. But this means that there exists additive $f\colon\real\to\real$ such that $F(x,y)=f(x)$ for every $x,y\in\real$. So, by Proposition~\ref{propA}, function $f$ is continuous. Thus $F\in\CC(\real^2)$. \qed \ex{ACnotD}{There exists an $f\in\Add(\real^2)\cap\AC(\real^2)\setminus\darb(\real^2)$.} \proof Let $\B$ be a family of all closed subsets $B$ of $\real^2\times\real$ for which the projection $\proj(B)$ onto first coordinate $\real^2$ has cardinality $\co$. (Thus, it contains a perfect set.) It is known that if $f\colon\real^2\to\real$ intersects every element of $\B$ then $f\in \AC(\real^2)$. (See \cite[prop.~1.2]{N1}.) Let $\{F_\xi\colon\xi<\co\}$ be an enumeration of $\B$. By induction on $\xi<\co$ define a sequence $\la \la a_\xi,b_\xi,y_\xi\ra\colon \xi<\co\ra$ such that for every $\xi<\co$ the following conditions hold. \begin{description} \item{(i)} $a_\xi\in\proj(B_\xi)\setminus \lin(\{a_\zeta\colon \zeta<\xi\}\cup\{b_\zeta\colon \zeta<\xi\})$. \item{(ii)} $b_\xi\in\proj(B_\xi)\setminus \lin(\{a_\zeta\colon \zeta\leq\xi\}\cup\{b_\zeta\colon \zeta<\xi\})$. \item{(iii)} $\la a_\xi,y_\xi\ra\in B_\xi$. \end{description} Let $A=\{a_\xi\colon\xi<\co\}$, $B=\{b_\xi\colon\xi<\co\}$, and $H$ be a Hamel base containing $A\cup B$. Define $f_0\colon H\to\real$ by putting $f_0(a_\xi)=y_\xi$ and $f_0(x)=1$ for $x\in H\setminus A$. Let $f\colon\real^2\to\real$ be a linear extension of $f_0$. Then $f$ is additive and almost continuous, since it intersects every $F\in\F$. However, it is not Darboux, since the set $D=B\cup\{0\}$ is connected, as it intersects every perfect subset of $\real^2$, while $f[D]=\{0,1\}$ is not connected. \qed \ex{DnotAC}{There exists an $f\in\Add(\real^2)\cap\darb(\real^2)\setminus\AC(\real^2)$.} \proof Take an additive Darboux function $g\colon\real\to\real$ with the property that \begin{description} \item{($*$)} $g[(a,b)]=\real$ for every $a0$. This follows from Fact~\ref{factContr}. \section{Some missing examples of additive Darboux-like functions on $\real$} \ex{ACandPRnotCIVP}{There exists an $f\in\Add\cap\AC\cap\PR\setminus\CIVP$. Moreover, $f[K]$ is not nowhere dense for every perfect set $K\subset\real$. } \proof Let $\P$ be a family of pairwise disjoint perfect sets such that the set $\bigcup \P$ is linearly independent and $|\{P\in\P\colon P\subset (a,b)\}|=\co$ for every $a