%Version of November 1, 2001; corrected to agree with Galley Proof \documentclass[12pt]{article} \usepackage{amssymb} \date{} \title{On sums of Darboux and nowhere constant continuous functions} \pagestyle{myheadings} \markboth{K.~Ciesielski and J.~Pawlokowski }{On sums of Darboux and nowhere constant continuous functions } \author{ Krzysztof Ciesielski\thanks{ AMS classification numbers: Primary 26A15; Secondary 03E35. \endgraf Key words and phrases: Darboux, nowhere constant, images of continuous functions. } \\ {\footnotesize Department of Mathematics,} {\footnotesize West Virginia University,} \\ {\footnotesize Morgantown, WV 26506-6310, USA}\\ {\footnotesize e-mail: K\_Cies@math.wvu.edu}; {\footnotesize web page: {\tt http://www.math.wvu.edu/\~{}kcies}} \and Janusz Pawlikowski\thanks{ The work of the second author was partially supported by KBN Grant 2 P03A 031~14.} \\ {\footnotesize Department of Mathematics,} {\footnotesize University of Wroc\l aw,} \\ {\footnotesize pl. Grunwaldzki 2/4, 50-384 Wroc\l aw, Poland;} {\footnotesize e-mail: pawlikow@math.uni.wroc.pl}\\ {\footnotesize and}\\ {\footnotesize Department of Mathematics,} {\footnotesize West Virginia University,}\\ {\footnotesize Morgantown, WV 26506-6310, USA} %{\footnotesize e-mail: pawlikow@math.wvu.edu} } \newcommand{\new}{\marginpar{{\tiny NEW}}} \newcommand{\ch}[1]{\marginpar{{\tiny #1}}} \newcounter{ChartNo} \newcommand{\Ccounter}{\refstepcounter{ChartNo}\theChartNo} \newcommand{\forces}{\mathrel{\|}\joinrel\mathrel{-}} \newcommand{\dec}{{\rm dec}} \newcommand{\ignore}[1]{} \newcommand{\IntTh}{{\rm IntTh}} \newcommand{\Implies}{\Longrightarrow} \newcommand{\SoIC}{{s_0^{\rm prism}}} \newcommand{\SoST}{{s_0^{\rm cube}}} \newcommand{\psm}{{\rm CPA}} \newcommand{\psmC}{\mbox{{\rm CPA$_{\rm cube}$}}} \newcommand{\psmP}{\mbox{{\rm CPA$_{\rm prism}$}}} \newcommand{\psmCsec}{\mbox{{\rm CPA$_{\rm cube}^{\rm sec}$}}} \newcommand{\psmPsec}{\mbox{{\rm CPA$_{\rm prism}^{\rm sec}$}}} 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\newtheorem{problem}[theorem]{Problem} \newtheorem{example}[theorem]{Example} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{Fact}[theorem]{Fact} \newtheorem{Claim}[theorem]{Claim} \newcommand{\thm}[2]{\begin{theorem}\label{#1}{\sl #2}\end{theorem}} \newcommand{\cor}[2]{\begin{corollary}\label{#1}{\sl #2}\end{corollary}} \newcommand{\prop}[2]{\begin{proposition}\label{#1}{\sl #2}\end{proposition}} \newcommand{\lem}[2]{\begin{lemma}\label{#1}{\sl #2}\end{lemma}} \newcommand{\pr}[2]{\begin{problem}\label{#1}{\rm #2}\end{problem}} \newcommand{\ex}[2]{\begin{example}\label{#1}{\sl #2}\end{example}} \newcommand{\defi}[2]{\begin{definition}\label{#1}{\rm #2}\end{definition}} \newcommand{\rem}[2]{\begin{remark}\label{#1}{\rm #2}\end{remark}} \newcommand{\fact}[2]{\begin{Fact}\label{#1}{\sl #2}\end{Fact}} \newcommand{\claim}[2]{\begin{Claim}\label{#1}{\sl #2}\end{Claim}} \begin{document} \maketitle \begin{abstract} We show that the property \begin{itemize} \item[(P)] 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{itemize} follows from the following two propositions: \begin{itemize} \item[(A)] for every subset $S$ of $\real$ of cardinality $\continuum$ there exists a uniformly continuous function $f\colon\R\to[0,1]$ such that $f[S]=[0,1]$, \item[(B)] for an arbitrary function $h\colon\real\to\real$ whose image $h[\real]$ contains a non-trivial interval there exists an $A\subset\real$ of cardinality $\continuum$ such that the restriction $h\restriction A$ of $h$ to $A$ is uniformly continuous, \end{itemize} which hold in the iterated perfect set model. \end{abstract} \newpage Our set theoretic terminology is standard and follows that of~\cite{CiBook}. In particular, functions are identified with their graphs and $|X|$ stands for the cardinality of a set $X$. The symbol $\continuum$ stands for $|\real|$ and $\C(\real)$ denotes the family of all continuous functions from $\real$ into $\real$. A function $f\colon\real\to\real$ is {\em Darboux}\/ if a conclusion of the intermediate value theorem holds for $f$ or, equivalently, when $f$ maps every interval onto an interval; $f$ is a {\em Sierpi\'nski-Zygmund function}\/ if its restriction $f\restriction Y$ is discontinuous for every subset $Y$ of $\real$ of cardinality $\continuum$; $f$ is {\em nowhere constant}\/ if it is not constant on any non-trivial interval. The class of Darboux functions has been studied for a long time as one of possible generalizations of the class of continuous functions. However, it has some peculiar properties. For example, it is not closed under addition. In fact, in 1927 Lindenbaum \cite{Li} noticed that every function $f\colon\real\to\real$ can be written as a sum of two Darboux functions (see also \cite{SMa}) while Erd\H{o}s~\cite{Erdos} showed that if additionally $f$ is measurable then both of the summands can be chosen to be measurable. In~1990 Kirchheim and Natkaniec showed that there are some truly bad Darboux functions by proving the following. \prop{pr:NatKir}{ {\rm (Kirchheim, Natkaniec~\cite{NK})} If the 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 $f\colon\R\to\R$.} 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. \prop{pr:Stepr1}{ {\rm (Stepr\=ans~\cite{St})} It is consistent with ZFC that \begin{itemize} \item[{\rm (P)}] 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{itemize} } In his paper Stepr\=ans showed that (P) holds in the iterated perfect set model. His direct forcing argument is quite complicated and difficult to follow. We will show that (P) follows from the principles (A) and (B) defined in the abstract. Since the proofs that (A) and (B) hold in the the iterated perfect set model are much simpler than the argument given in~\cite{St}, our argument for (P) is essentially more accessible. The proof that (A) holds in the iterated perfect set model is due to Miller~\cite{Mi}. The proof that (B) holds in this model is due to Balcerzak, Ciesielski, and Natkaniec~\cite{BCN} and consists of a page long argument based on the results from~\cite{Mi}. In addition, properties (A) and (B) follow quite easily from the Covering Property Axiom CPA as shown by the authors in~\cite{CPAbook}. For more on the Darboux property see also the survey article~\cite{KCsurv}. Note also that condition (B) is equivalent to its following seemingly stronger version: \begin{itemize} \item[(B$^*$)] for an arbitrary $S\subset\real$ and a function $h\colon S\to\real$ whose image $h[S]$ contains a non-trivial interval there exists an $A\subset S$ of cardinality $\continuum$ such that the restriction $h\restriction A$ of $h$ to $A$ is uniformly continuous. \end{itemize} Clearly (B$^*$) implies (B). To see the other implication let $h$ be as in (B$^*$) and $f\colon\real\to\real$ be Sierpi\'nski-Zygmund. Put $h^*=f\restriction(\real\setminus S)\cup h$ and apply (B) to find $A^*\subset\real$ of cardinality $\continuum$ such that $h^*\restriction A^*$ is uniformly continuous. Then $A=A^*\cap S$ is as in (B$^*$). Our main theorem is the following. \thm{thDarb}{ Assume that {\rm (A)} and {\rm (B)} hold. Then for every Darboux function $d\colon\real\to\real$ there exist a complete metric $\rho$ on $\C(\real)$ and a dense $G_\delta$ subset $G$ of $\la \C(\real),\rho\ra$ of nowhere constant functions such that $d+g$ is Darboux for every $g\in G$. In particular {\rm (A)}\&{\rm (B)} implies {\rm (P)}.} Our proof will be based on the following two lemmas. \lem{lemDarb}{ Let $d\colon\real\to\real$ be Darboux and $D\subset\real$ be such that $d\restriction D$ is dense in $d$. If $g\colon\real\to\real$ is continuous and such that \begin{equation}\label{eqD1} [(d+g)(\alpha),(d+g)(\beta)]\subset (d+g)[[\alpha,\beta]] \ \ \mbox{ for all \ \ $\alpha,\beta\in D$} \end{equation} then $d+g$ is Darboux. } \proof An easy proof can be found in~\cite[lemma~4.1]{St}. \qed \ignore{ \lem{lemDarb1}{If $d\colon\real\to\real$ is Darboux and $f\colon\real\to\real$ is continuous then for every $\alpha<\beta$ the set $[(d+f)(\alpha),(d+f)(\beta)]$ is dense in $(d+f)[[\alpha,\beta]]$. } \proof {\bf Give a reference or add an easy proof.} } \lem{lemDarb2a}{Assume that {\rm (A)} holds and for every $n<\omega$ let $A_n\in[\real]^\continuum$. Then for every $n<\omega$ there exists a $B_n\in[A_n]^\continuum$ such that the closures of $B_n$'s are pairwise disjoint. } \proof First note that (A) implies that \begin{equation}\label{eqDarb2} \forall \la C_n\in[\real]^\continuum\colon n<\omega\ra \ \exists C\in[C_0]^\continuum \ \forall n<\omega \ |C_n\setminus\cl(C)|=\continuum. \end{equation} Indeed, by condition (A), we can find a continuous function $f\colon\real\to[0,1]$ such that $|C_0\cap f^{-1}(c)|=\continuum$ for every $c\in[0,1]$. (Just take a composition of a function from (A) with the Peano curve followed by a projection.) Identify $2^\omega$ with a subset of $[0,1]$ and by induction on $n<\omega$ choose an increasing sequence $s_n\in 2^{n+1}$ such that $|C_n\setminus f^{-1}(\{c\in2^\omega\colon s_n\subset c\})|=\continuum.$ Put $s=\bigcup_{n<\omega}s_n$. Then $C=C_0\cap f^{-1}(s)$ satisfies (\ref{eqDarb2}). Next note that there exists a sequence $\la D_n\colon n<\omega\ra$ of closed subsets of $\real$ such that $\left|A_n\cap\left(D_n\setminus\bigcup_{i0$. Let $B(f,\e)$ be the open $\rho$-ball centered at $f$ and of radius $\e$. We will find $g\in B(f,\e)$ and $\delta>0$ such that $B(g,\delta)\subset B(f,\e)\cap H_\alpha^\beta$. So take an $n<\omega$ such that $2^{-n}<\e/4$, choose a non-empty open interval $J\subset (\alpha,\beta)\setminus\bigcup_{i=1}^n A_i$, and pick different $x,y\in J$. It is easy to find $g\in \C(\real)$ such that $g(x)\neq g(y)$, $g\restriction (\real\setminus J)=f\restriction (\real\setminus J)$, and $\rho_0(f,g)<2^{-n}$. Then, by the choice of $J$ we also have $\rho_1(f,g)<2^{-n}$, so $g\in B(f,\e/4)$. Now, if $\delta=\min\{|g(x)-g(y)|/4,\e/4\}$ then $B(g,\delta)\subset B(f,\e)\cap H_\alpha^\beta$. To see that each $G_\alpha^\beta$ contains a dense open subset take $f\in \C(\real)$ and $\e>0$. As previously we will find $g\in B(f,\e)$ and $\delta>0$ such that $B(g,\delta)\subset B(f,\e)\cap G_\alpha^\beta$. Find $\alpha=x_0n$ such that $d\restriction A_{k_i}\subset(x_i,x_{i+1})\times[d(x_i),d(x_{i+1})]$. By (A) for every $i