\documentclass{rae} \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsthm} \MathReviews{Primary 26A15; Secondary 54C30.} \keywords{Darboux functions, extendable functions, almost continuous functions, connectivity functions, functions with perfect road, peripherally continuous functions, CIVP-functions, SCIVP-functions, quasi-continuous functions} \markboth{R. G. Gibson and T. Natkaniec}{Darboux Like Functions} \author{Richard G. Gibson, Department of Mathematics, Columbus State University, Columbus, Georgia, 31907, USA, email: gibson\_richard@colstate.edu\\ Tomasz Natkaniec\footnote{This work was partially supported by NSF Cooperative Research Grant INT-9600548 with its Polish part financed by KBN. The final version of the paper was prepared during a visit of the second author at Columbus State University and West Virginia University.}, Department of Mathematics, Gda{\'n}sk University, Wita Stwosza 57, 80-952 Gda{\'n}sk, Poland, email: mattn@ksinet.univ.gda.pl} \title{DARBOUX LIKE FUNCTIONS. OLD~PROBLEMS~AND~NEW~RESULTS} \newcommand{\F}{{\cal F}} \def\P{{\cal P}} \newcommand{\A}{{\cal A}} \newcommand{\B}{{\cal B}} \newcommand{\C}{{\cal C}} \newcommand{\K}{{\cal K}} \newcommand{\CAB}{{\C_{\A,\B}}} \newcommand{\DAB}{{\C^{-1}_{\A,\B}}} \newcommand{\ACS}{{\rm ACS}} \newcommand{\Const}{{\rm Const}} \newcommand{\G}{{\cal G}} \newcommand{\CC}{{\rm C}} \newcommand{\PR}{{\rm PR}} \newcommand{\PC}{{\rm PC}} \newcommand{\PB}{{\rm PB}} \newcommand{\PI}{{\rm PI}} \newcommand{\D}{{\rm D}} \newcommand{\Conn}{{\rm Conn}} \newcommand{\CIVP}{{\rm CIVP}} \newcommand{\SCIVP}{{\rm SCIVP}} \newcommand{\WCIVP}{{\rm WCIVP}} \newcommand{\DIVP}{{\rm DIVP}} \newcommand{\QU}{{\rm QU}} \newcommand{\Ext}{{\rm Ext}} \newcommand{\Q}{{\rm Q}} \newtheorem{Th}{Theorem} \newtheorem{Le}[Th]{Lemma} \newtheorem{Co}[Th]{Corollary} \newtheorem{Po}[Th]{Proposition} \newtheorem{Ro}[Th]{Remark} \newtheorem{Qu}[Th]{Question} \newtheorem{Pro}[Th]{Problem} \newcommand\mathN{{\mathbb N}} \newcommand\mathI{{\mathbb I}} \newcommand\mathR{{\mathbb R}} \newcommand\mathQ{{\mathbb Q}} \newcommand\real{\mathR} \newcommand{\RRR}{\mathR^{\mathR}} \newcommand{\restr}{{\restriction}} \newcommand{\co}{{\mathfrak c}} \newcommand\id{{\rm id}} \newcommand\cl{{\rm cl}\,} \newcommand\Cl{{\rm cl}\,} \newcommand\rng{{\rm rng}\,} \newcommand\dom{{\rm dom}\,} \newcommand\bd{{\rm bd}\,} \newcommand\Int{{\rm int}\,} \newcommand{\pf}{{\noindent\sc Proof. }} \newcommand{\charf}[1]{\mbox{\raise.48ex\hbox{$\chi$}$_{#1}$}} \begin{document} \maketitle During the 14th Summer School on Real Functions Theory in Liptovsky J{\'a}n, Slovakia in 1996, Richard G. (Jerry) Gibson gave a talk concerning Darboux-like functions. Following his talk we wrote the survey article {\it Darboux-like functions}~\cite{GN}, that was published in the {\it Real Analysis Exchange}. In that paper we collected not only known facts on the subject but also presented some open problems. In the last two years a considerable numbers of papers have been written that solve some of the open questions posed in our survey article. In this short abstract we would like to present those solutions and to list the problems that remain open. It is based on the talk given by Tomasz (Tomek) Natkaniec at the 15th Summer School on Real Functions Theory in Liptovsky J{\'a}n, Slovakia, in 1998. Most of the new results are contained as a part of the Polish-American project {\it Set Theoretic Analysis} and has been described in K.~Ciesielski's survey {\it Set Theoretic Real Analysis}~\cite{C}. You can find these results on the Set Theoretic Analysis web page: \centerline{http://www.math.wvu.edu/homepages/kcies/STA/STA.html} \section{Notations and definitions} We use the notations from our survey article~\cite{GN}. In particular, the questions in this paper are numbered the same as in the survey article. Recall the main notations and definitions from the survey article. \begin{description} \item[D] -- $f$ is a {\it Darboux function} if $f(C)$ is connected whenever $C$ is connected in $X$; \item[Conn] -- $f$ is a {\it connectivity function} if the graph of $f$ restricted to $C$ is connected in $X\times Y$ whenever $C\subset X$ is connected; \item[ACS] -- $f$ is an {\it almost continuous function} in the sense of Stallings, if $U$ is an open subset of $X\times Y$ containing the graph of $f$, then $U$ contains the graph of a continuous function $g:X\to Y$; \item[Ext] -- $f$ is an {\it extendable function} if there exists a connectivity function $g:X\times I\to Y$ such that $f(x)=g(x,0)$ for all $x\in X$. \end{description} \section{Borel measurable functions} \medskip It is well known that in the class of all functions from $\mathR$ into $\mathR$ the following chain of implications holds: $$ (\star)\;\;\;\Ext \rightarrow\ACS \rightarrow\Conn \rightarrow\D $$ For Baire class 1 functions, $f\colon\mathR\to\mathR$, the properties defined above are equivalent, i.e., $$\Ext=\ACS=\Conn=\D.$$ However, for Borel measurable functions, Brown, Humke and Laczkovich proved that the implications $(\star)$ are not reversible except for possibly $\Ext \rightarrow\ACS$~\cite{BHL}. Thus they posed the following question. \medskip \noindent {\bf Question~3.21. }{\em Does there exist a Borel function $f\colon\mathR\to \mathR$ such that $f\in\ACS\setminus\Ext$?} \medskip The answer to this question was given in the following theorem. \medskip \begin{Th}\label{th1} {\rm (Ciesielski, Jastrz{\c{e}}bski, 1998) \cite{CJ}} $$(\ACS\setminus \Ext)\cap B_2\neq \emptyset.$$ \end{Th} This answers also {\bf Question~3.11}. (See the next section.) \medskip J.~Brown~\cite{JB} has considered whether the implications $(\star)$ within the class of all functions with $G_{\delta}$ graphs are reversible. He noticed that in this class all implications $\ACS \rightarrow\Conn \rightarrow\D$ are not reversible. (Recall that every function in the class $B_1$ has a graph which is a $G_{\delta}$ subset of $\mathR^2$. Moreover, it is well-known that every function with a $G_{\delta}$ graph is Borel measurable. However, it can be of an arbitrary high Borel class. See e.g., \cite{AK}.) The problem of whether the implication $\Ext\rightarrow\ACS$ in the class of all functions with a $G_{\delta}$ graph is reversible was posed by R.~Gibson~\cite{RG}. \medskip \noindent {\bf Question~3.15. }{\em Does there exist a function $f\in\ACS\setminus\Ext$ with a $G_{\delta}$ graph?} \medskip This question was answered in the next theorem. \medskip \begin{Th} {\rm (Ciesielski, Rosen, 1999) \cite{CRos} There exists a Baire two function $f\in\ACS\setminus\Ext$ with a $G_{\delta}$ graph.} \end{Th} The next problem is connected with the \underline{Ces{\'a}ro-Vietoris function} $\varphi:{\bf I} \to {\bf I}$ defined by the formula: $$\varphi(x)= \overline{\lim}_{n \to\infty}\frac{a_{1}+ \ldots +a_{n}}{n}$$ where $a_{i}$ are given by the unique nonterminating binary expansion of the number $x=(0.a_{1}a_{2} \ldots )$. \medskip \noindent {\bf Claims. } \begin{itemize} \item $\varphi\in B_2$. \item {\em (Vietoris, 1921). $\varphi\in\Conn$.} \item {\em (Brown, 1975). $\varphi\in\ACS$.} \end{itemize} \medskip \noindent {\bf Question~3.22. }{\em Does the Ces{\'a}ro-Vietoris function belong to $\Ext$?} {\bf Open.} \section{Cantor intermediate values properties} Recall the following definitions. \begin{description} \item[PR] -- $f$ has a {\it perfect road} if for every $x\in \mathR$, there exists a perfect set $P$ having $x$ as a bilateral limit point such that $f\restr P$ is continuous at $x$; \item[CIVP] -- {\it Cantor Intermediate Value Property}: $f\in\CIVP$ if for all $p,q\in \mathR$ with $p\neq q$ and $f(p)\neq f(q)$ and for every Cantor set $K$ between $f(p)$ and $f(q)$, there exists a Cantor set $C$ between $p$ and $q$ such that $f(C)\subset K$; \item[SCIVP] -- {\it Strong Cantor Intermediate Value Property}: $f\in\SCIVP$ if for all $p,q\in\mathR$ with $p\neq q$ and $f(p)\neq f(q)$ and for every Cantor set $K$ between $f(p)$ and $f(q)$, there exists a Cantor set $C$ between $p$ and $q$ such that $f(C)\subset K$ and $f\restr C$ is continuous. \item[PC] -- $f$ is {\it peripherally continuous} if for every $x\in X$ and for all pairs of open sets $U$ and $V$ containing $x$ and $f(x)$, respectively, there exists an open subset $W\subset U$ such that $x\in W$ and $f(\bd(W))\subset V$, where $\bd(W)$ denotes the boundary of $W$. \end{description} \medskip For Baire class 1 functions, $f\colon\mathR\to\mathR$, the functions defined above are equivalent, i.e., $$\Ext=\ACS=\Conn=\D=\SCIVP=\CIVP=\PR=\PC.$$ \medskip For arbitrary function, $f\colon\mathR\to\mathR$, we have only the following implications \begin{picture}(0,95) \put(30,50){\makebox(0,0){$\Ext$}} \put(40,52){\vector(1,1){20}} \put(40,48){\vector(1,-1){20}} \put(80,72){\makebox(0,0){$\ACS$}} \put(85,30){\makebox(0,0){$\SCIVP$}} \put(95,72){\vector(1,0){35}} \put(148,72){\makebox(0,0){$\Conn$}} \put(165,72){\vector(1,0){35}} \put(210,72){\makebox(0,0){$\D$}} \put(110,30){\vector(1,0){20}} \put(155,30){\makebox(0,0){$\CIVP$}} \put(175,30){\vector(1,0){20}} \put(210,30){\makebox(0,0){$\PR$}} \put(225,30){\vector(1,1){20}} \put(225,72){\vector(1,-1){20}} \put(255,50){\makebox(0,0){$\PC$}} \end{picture} \begin{center} Chart 1 \end{center} \medskip The following theorems and questions are related to Chart 1. \medskip \begin{Th} {\rm (K. Banaszewski, Natkaniec, 1996)~\cite{BN}} Assume CH. Then $(\ACS\cap\CIVP)\setminus\Ext\neq\emptyset$. \end{Th} \medskip \noindent {\bf Question. }{\em Can the above be proved in ZFC?} \begin{Th} {\rm (Ciesielski, 1997)~\cite{KC}} Yes. \end{Th} \medskip \noindent {\bf Question~3.11. }{\em Does there exist $f\in (\ACS\cap\SCIVP)\setminus\Ext?$} \begin{Th} {\rm (Rosen, 1997)~\cite{HR}} Yes, under CH. \end{Th} \begin{Th} {\rm (Ciesielski, Ros{\l}anowski, 1998)~\cite{CR}} Every $g\in\Ext$ with a dense graph satisfies the following condition: \begin{description} \item[SSCIVP] -- for all $a1$. \end{Th} \section{Darboux like functions and quasi-continuity} In general Darboux like functions and quasi-continuous functions are not related. However, under certain conditions quasi-continuous functions are extendable. One of these conditions is that the function have a graph whose closure is bilaterally dense in itself. (It means that the right and left cluster sets of $f$ at any point $x$ of the domain of $f$ coincide.) \begin{description} \item[QC] --$f:X\to Y$ is a quasi-continuous function if and only if for each $p\in X$ the following condition holds: for every open set $U\subset X$ with $p\in U$ and open set $V\subset Y$ with $f(p)\in V$ there exist a non-empty open set $W\subset U$ such that $f(W)\subset V$. \end{description} \begin{Th} {\rm (Rosen, 1998)~\cite{HR2}} If $f$ is Darboux, quasi-continuous and has a graph whose closure is bilaterally dense in itself, then $f$ is extendable and $D(f)$ is $f$-negligible. \end{Th} (Here $D(f)$ denotes the set of all points at which $f$ is discontinuous.) Recently, Francis Jordan \cite{FJ} improved this results by showing that we need not assume that the function is a Darboux function. \section{Functions of $n>1$ variables} For $n>1$, the implications in Chart 1 are no longer valid. In fact, we have the followig diagram. \begin{picture}(0,95) \put(60,50){\makebox(0,0){$\Ext$}} \put(80,51){\vector(1,0){20}} \put(135,50){\makebox(0,0){$\Conn=\PC$}} \put(170,52){\vector(1,1){20}} \put(170,48){\vector(1,-1){20}} \put(210,72){\makebox(0,0){$\ACS$}} \put(210,30){\makebox(0,0){$\D$}} \end{picture} \medskip \noindent {\bf Question~8.3. }{\em Is the inclusion $\Ext\subset\Conn$ proper for $f\colon\mathR^n\to\mathR$, if $n>1$?} \begin{Th} {\rm (Ciesielski, Natkaniec, Wojciechowski, 1998)~\cite{CNW}} Every connectivity function $f\colon\mathR^n\to\mathR$, $n>1$ can be extended to a connectivity function of $n+1$ variables. \end{Th} Thus for functions of $n>1$ variables we have the equalities $$\Ext=\Conn=\PC.$$ \section{Compositions} \medskip This section contains the new results concerning compositions of Darboux like functions. \medskip \noindent {\bf Question~9.3. }{\em Is every Darboux function the composition of two (finitely many) $\ACS$ (or $\Conn$) functions?} \begin{Th} {\rm (Natkaniec, 1991)~\cite{TN}} Assume CH. Every $f\in\D$ with dense level sets is the composition of two $\ACS$ functions. \end{Th} \medskip \noindent {\bf Question. }{\em Can the above be proved in ZFC?} {\bf Open.} \begin{Th} {\rm (Kellum, 1998)~\cite{KK}} There exists an $f\in\Conn$ such that $f$ is not the composition of finitely many $\ACS$ functions. \end{Th} \begin{Th} {\rm (Ciesielski, Kellum, 1998)~\cite{CK}} There exists an $f\in\D$ such that $f$ is not the composition of finitely many $\Conn$ functions. \end{Th} \noindent {\bf Question~9.1. }{\em If $f,g\in\Ext$, is $g\circ f\in\Ext$?} {\bf Open.} \medskip The next result seems to be connected with the above problem. (Note that every $B_2$ function can be expressed as the composition of two $B_1$ functions.\footnote{This fact was proved for us by S.~Solecki.}) \medskip \noindent {\bf Question. }{\em Is every $f\in\D B_2$ the composition of two derivatives ($\D B_1$ functions)?} {\bf No. }See Kellum's example in~\cite{KK}. So the folowing problems seems to be {\bf open}. \medskip \noindent {\bf Question.} (Ciesielski) {\rm Has the compositions of two derivatives ($\D B_1$ or $\Ext$) functions from $\mathI$ into $\mathI$ a fix point?} (Recall that every connectivity function from $\mathI$ into $\mathI$ (so also every derivative, every $\D B_1$ and $\Ext$ function) has fixed points.) \medskip \noindent {\bf Question. }{\em Characterize compositions of of two derivatives ($\D B_1$ functions).} \section{Uniform limits} Let $\overline{\F}$ denote the class of uniform limits of sequences of functions from $\F$. \medskip \noindent {\bf Question 9.13. }{\em Does there exists $f\in\ACS\cap\PR\setminus\overline{\Ext}$?} \medskip The answer is affirmative. Both Rosen's function from~\cite{HR} (constructed under CH) and ZFC Ciesielski-Jastrz{\c{e}}bski's example from~\cite{CJ} are in the class $\ACS\cap\SCIVP$ but are not in $\overline{\Ext}$. \medskip \noindent {\bf Question 9.14. }{\em Characterize the uniform limits of sequences of $\Ext$ functions ($\ACS$ functions or $\Conn$ functions).} \medskip This problem is still {\bf open}. However, the following conjecture was formulated by K.~Kellum during the last Miniconference in Real Analysis (March 1999) at Auburn University. \medskip \noindent {\bf Conjecture.} {\em Let $f\colon\mathR\to\mathR$ be bounded. Then $f\in\overline{\rm ACS}$ iff \begin{itemize} \item $f\in {\rm U}\; (=\overline{\rm D})$; \item $f$ is away-almost continuous: \end{itemize}} A function $f\colon\mathR\to\mathR$ is {\it away almost continuous} if for every closed set $B\subset\mathR^2$ such that $\sup\{ {\rm dist}(f(x),B_x)\colon \; x\in \mathR\}>0$ there exists a continuous function $g\colon\mathR\to\mathR$ which is disjoint with $B$. \section{Universal summands} It is easy to see that there is no $g\in B_{\alpha}$ such that $B_{\alpha}+g\subset\D$. Similarly, there is no Borel measurable function $g$ such that $f+g\in\D$ for all Borel functions $f$. \medskip \noindent {\bf Question.}{ \em {\rm (Ceder, 1965) \cite{JC}} Does there exist a Borel measurable $g$ such that $B_{\alpha}+g\subset {\rm D}$?} \begin{Th} {\rm (Natkaniec, Rec{\l}aw, 1997)~\cite{NR}} For each $\alpha < \omega_1$ there is a Borel function with $B_{\alpha}+g\subset {\rm ACS}$. \end{Th} \medskip \noindent {\bf Question. }{\em Does there exist $g\in B_{\alpha+1}$ such that $B_{\alpha}+g\in {\rm D}$ ($B_{\alpha}+g\in {\rm ACS}$ or $B_{\alpha}+g\in {\rm Ext}$)?} \begin{Th} {\rm (Solecki, 1998)~\cite{SS}} For each $\alpha<\omega_1$ there is a $g\in B_{\alpha}$ such that $\bigcup_{\beta<\alpha}B_{\beta}+g\in \Ext$. \end{Th} For an ordinal $\alpha<\omega_1$ define the cardinal $$u(\alpha)=\min(\{ |{\cal F}|\colon\; {\cal F}\subset B_{\alpha}\;\&\; (\forall\; g\in B_{\alpha})\; ({\cal F}+g\not\subset \D)\}\cup\{ \co^+\})$$ Clearly, $u(0)=\co^+$. Moreover, $u(1)=\omega$~\cite{PP} and $\omega1$ (an easy corollary from \cite{JC}). Thus we have the following \begin{Th} Under CH, $u(\alpha)=\omega_1$ for all $\alpha>1$. \end{Th} \medskip \noindent {\bf Question. }{\em Can the above be proved in ZFC?} {\bf Open.}\footnote{In fact, this is a reformulation of Ceder's problem from~\cite{JC}.} \section{Products} \begin{Th} For every $f\colon\mathR\to\mathR$ the following conditions are equivalent: \begin{itemize} \item $f$ is a product of two $\D$ functions; \item $f$ is a product of two $\Conn$ functions; \item $f$ is a product of two $\ACS$ functions; \item $f$ possesses the (JC) property: \begin{quote} $f$ has a zero in each subinterval in which it changes sign. \end{quote} \end{itemize} \end{Th} \medskip \noindent {\bf Question~9.31. }{\em Is every $f\colon\mathR\to\mathR$ with (JC) property the product of two $\Ext$ functions?} {\bf Open.} \begin{thebibliography}{10} \bibitem[BN]{BN} K.~Banaszewski, T.~Natkaniec, {\em Sierpi{\'n}ski-Zygmund functions that have the Cantor intermediate value property}, Real Anal. 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