% Some examples concerning \I-density continuous functions \def\change{\vrule height10pt width2pt depth2pt} %\documentstyle[12pt,rae]{article} \documentclass{rae} \pagestyle{myheadings} %\input{exampmacros} %%%%%%%%%%%%%%%%%%%%% HYPHENATIONS \hyphenation{ap-prox-im-ate-ly} \hyphenation{dif-fer-ent-iab-le} \hyphenation{den-sity} \hyphenation{con-tin-uous} \hyphenation{func-tion} \hyphenation{dis-per-sion} \hyphenation{di-ver-gent} %%%%%%%%%%%%%%%%%%%%% SUBSETS OF THE REALS %sets with the Baire property \newcommand{\BaireSets}{\mbox{$\cal B$}} %Lebesgue measurable sets \newcommand{\Lebesgue}{\mbox{$\cal L$}} % Use blackboard bold for the normal real subsets \newfont{\amssymten}{msbm10 scaled\magstep1} \newfont{\amssymnine}{msbm9} \newfont{\amssymbig}{msbm10 scaled\magstep2} % natural numbers \newcommand{\nats}{\mbox{{\amssymten N}}} %Mac %Next a natural number sumbol for super and sub \newcommand{\smallnats}{\mbox{{\amssymnine N}}} %Mac % rational numbers \newcommand{\rationals}{\mbox{{\amssymten Q}}} %Mac % real numbers \newcommand{\reals}{\mbox{{\amssymten R}}} %Mac \newcommand{\smallreals}{\mbox{{\amssymnine R}}} %%%%%%%%%%%%%%%%%%%%% IDEALS %the ideal of first category sets \newcommand{\I}{\mbox{$\cal I$}} %the ideal of measure zero sets \newcommand{\N}{\mbox{$\cal N$}} %generic ideal \newcommand{\J}{\mbox{$\cal J$}} %sets with cardinality < continuum \newcommand{\lessthanc}{\mbox{${\cal I}_c$}} %empty ideal \newcommand{\emptyideal}{\mbox{$\cal 0$}} %nowhere dense sets \newcommand{\nwdense}{\mbox{$\I_0$}} %%%%%%%%%%%%%%%%%%%%%% FUNCTION OPERATIONS %inverse of a function \newcommand{\inv}[1]{\mbox{$#1^{-1}$}} %reals to reals \newcommand{\RtoR}{\mbox{$\colon\reals\to\reals$}} %conjugation \newcommand{\conj}[2]{\mbox{$#2\circ #1\circ\inv{#2}$}} %oscillation \newcommand{\osc}{\omega} %restriction \newcommand{\fP}{\mbox{$f|_P$}} %support \newcommand{\supp}[1]{\mbox{supp$#1$}} %%%%%%%%%%%%%%%%%%%%%% SET OPERATIONS %interior of a set \newcommand{\interior}[1]{\mbox{$\mbox{{\rm int}}\left(#1\right)$}} %closure of a set \newcommand{\closure}[1]{\mbox{$\mbox{\rm cl}\left(#1\right)$}} %complement of a set \newcommand{\complement}[1]{\mbox{$#1^c$}} % big union \newcommand{\Cup}{\bigcup} % big intersection \newcommand{\Cap}{\bigcap} % union over the natural numbers \newcommand{\UoverN}{\Cup_{n\in\smallnats}} % characteristic function \newcommand{\charf}[1]{\mbox{\raise.48ex\hbox{$\chi$}$_{#1}$}} % distance between sets \newcommand{\dist}[2]{\mbox{dist$(#1,#2)$}} %cardinality of a set \newcommand{\card}[1]{\mbox{$\mbox{card}\left(#1\right)$}} %measure of a set \newcommand{\measure}[1]{\mbox{$\mbox{m}(#1)$}} \newcommand{\innermeasure}[1]{\mbox{$\mbox{m}_i(#1)$}} \newcommand{\outermeasure}[1]{\mbox{$\mbox{m}_o(#1)$}} %density points \newcommand{\idensitypts}[1]{\mbox{$\Phi_{\cal I}(#1)$}} \newcommand{\deepidensitypts}[1]{\mbox{$\Phi_{\cal D}(#1)$}} \newcommand{\jdensitypts}[2]{\mbox{$\Phi_{\cal #1}(#2)$}} \newcommand{\densitypts}[1]{\mbox{$\Phi_{\cal N}(#1)$}} %%%%%%%%%%%%%%%%%%%%%% THEOREM-LIKE THINGS %Note the capitals to differentiate them from the commands. \newtheorem{Theorem}{Theorem}[section] \newtheorem{Observation}[Theorem]{Observation} \newtheorem{Lemma}[Theorem]{Lemma} \newtheorem{Corollary}[Theorem]{Corollary} \newtheorem{Proposition}[Theorem]{Proposition} \newtheorem{Example}[Theorem]{Example} \newtheorem{Problem}{Problem}[section] %define shorter commands for these environments \newcommand{\theorem}[2]{\begin{Theorem}\Label{#1}$\!\!\!${\bf.} #2\end{Theorem}} \newcommand{\lemma}[2]{\begin{Lemma}\Label{#1}$\!\!\!${\bf.} #2\end{Lemma}} \newcommand{\corollary}[2]{\begin{Corollary}\Label{#1}$\!\!\!${\bf.} #2\end{Corollary}} \newcommand{\example}[2]{\begin{Example}\Label{#1}$\!\!\!${\bf.} #2\end{Example}} \newcommand{\proposition}[2]{\begin{Proposition}\Label{#1}$\!\!\!${\bf.} #2\end{Proposition}} \newcommand{\observation}[2]{\begin{Observation}\Label{#1}$\!\!\!${\bf.} #2\end{Observation}} \newcommand{\problem}[2]{\begin{Problem}\Label{#1}$\!\!\!${\bf.}#2\end{Problem}} %%%%%%%%%%%%%%%%%%%%%% LABEL MACRO %Macro to include theorem labels in the margins. \newcommand{\marginlabels}[1]{\def\margintags{#1}} \marginlabels{n} \def\nolabels{n} \newcommand{\Label}[1] {\if\margintags\nolabels \label{#1} \else \label{#1}\marginpar{{\tiny #1}} \fi} %%%%%%%%%%%%%%%%%%%%% MISCELLANEOUS \newcommand{\e}{\mbox{$\varepsilon$}} \renewcommand{\d}{\mbox{$\delta$}} \newcommand{\iapp}{\I-approximately} \newcommand{\idens}{\I-density} \newcommand{\idisp}{\I-dispersion} \hyphenation{-appro-xi-ma-te-ly} \hyphenation{-dif-fer-ent-iable} \newcommand{\io}{\mbox{I_0}} \newcommand{\sumi}[2]{\mbox{$\sum_{i=1}^{#1} #2$}} \newcommand{\F}{\mbox{$\cal F$}} \renewcommand{\H}{\mbox{$\cal H$}} \newcommand{\M}{\mbox{$\cal M$}} \newcommand{\D}{\mbox{$\cal D$}} \newcommand{\R}{\mbox{$\cal R$}} \renewcommand{\O}{\mbox{$\cal O$}} \newcommand{\Gdelta}{\mbox{$\mbox{\bf G}_\delta$}} \newcommand{\Y}{\mbox{$\Psi$}} \newcommand{\K}{\mbox{$\cal K$}} %%%%%%%%%%%%%%%%%%%%% FUNCTION SPACES \newcommand{\C}{\mbox{$\cal C$}} %continuous functions \newcommand{\ordcont}{\mbox{$\C_{\cal OO}$}} %approximately continuous functions \newcommand{\appr}{\mbox{$\C_{\cal NO}$}} %density continuous functions \newcommand{\denscont}{\mbox{$\C_{\cal NN}$}} %I-approx cont funct \newcommand{\iappr}{\mbox{$\C_{\cal IO}$}} % deep-I-approx cont funct \newcommand{\deepiappr}{\mbox{$\C_{\cal DO}$}} %I-dens cont funct \newcommand{\idenscont}{\mbox{$\C_{\cal II}$}} %deep I-dens cont funct \newcommand{\deepidenscont}{\mbox{$\C_{\cal DD}$}} % Baire one functions \newcommand{\dbone}{\mbox{${\cal DB}_1$}} % Baire star functions \newcommand{\dbstar}{\mbox{${\cal DB}_1^*$}} % Infinitely differentiable functions \newcommand{\cinfinity}{\mbox{$\C^\infty$}} % Constant functions \newcommand{\const}{\mbox{$Const$}} %%%%%%%%%%%%%%%%%%% TOPOLOGIES \newcommand{\T}{\mbox{$\cal T$}} %ordinary topology on R \newcommand{\ordinarytop}{\mbox{$\T_{\cal O}$}} %ordinary density topology on R \newcommand{\densitytop}{\mbox{$\T_{\cal N}$}} %I-density topology \newcommand{\itopology}{\mbox{$\T_{\cal I}$}} %Zaicek p*-topology (I-density topology) \newcommand{\pstar}{\mbox{${\cal P}^*$}} %J-density topology witch phi \newcommand{\phitopology}[1]{\mbox{$\T^\prime_{\cal #1}$}} %J-density topology witch psi \newcommand{\psitopology}[1]{\mbox{$\T^{\prime\prime}_{\cal #1}$}} %deep I-density topology \newcommand{\deepitopology}{\mbox{$\T_{\cal D}$}} \markright{Various continuities} \marginlabels{n} \title{ Various continuities with the density, \idens\ and ordinary topologies on {\amssymbig R}} \date{} \begin{document} \author{Krzysztof Ciesielski, Department of Mathematics, West Virginia University, Morgantown, WV 26506\footnote{Partially supported by a West Virginia University research grant.} \and Lee Larson, Department of Mathematics, University of Louisville, Louisville, KY 40292\footnote{This author was given support by a University of Louisville College of Arts and Sciences grant.}} \maketitle \section{\hskip -1em{.} Preliminaries} This paper is concerned with four topologies on the real line: the ordinary topology, density topology, \idens\ topology and deep-\idens\ topology. Using different topologies on the domain and range, these four topologies determine sixteen different ways a real function can be continuous. These continuities are examined and their relationships are determined. In addition, they are compared with the classes of analytic, \cinfinity, Baire*1, and Baire~1 functions. The notation used throughout this paper is standard. In particular, \reals\ stands for the set of real numbers and $\nats=\{1,2,3,\ldots\}$. The symmetric difference of the sets $A$ and $B$ is denoted by $A\triangle B$ and the complement of $A\subset\reals$ by \complement{A}. The symbols \Lebesgue\ and \BaireSets\ stand for the families of subsets of \reals\ which are Lebesgue measurable and have the Baire property, respectively. \N\ and \I\ denote the ideals of Lebesgue measure zero and first category subsets of \reals. If $A\in\Lebesgue$, then its Lebesgue measure is denoted by \measure{A}. It is easy to see that for any set $B\in\BaireSets$ there is a unique open set $\tilde{B}$ such that $\tilde{B}=\interior{\closure{\tilde{B}}}$ and $B=\tilde{B}\triangle I$ for some $I\in\I$. Any open set $G$ for which $G=\interior{\closure{G}}$ is called a {\em regular} open set. In a sense, $\tilde{B}$ is the largest open set such that $B$ can be written as $\tilde{B}\triangle I$ for some $I\in\I$. The symbol \dist{A}{B}\ stands for the Euclidean distance between the sets $A$ and $B$; i.e., $\dist{A}{B} = \inf\{|x - y|: x\in A, y\in B\}$. For $a\in\reals$ and $E\subset\reals$, define $E - a = \{x - a\in\reals\colon x\in E\}$ and $aE =\{ax\in\reals\colon x\in E\}$. The symbols \dbone, $\cal C$, \cinfinity\ and $\cal A$ stand for the classes of Darboux Baire one, ordinary continuous, infinitely many times differentiable and analytic functions from \reals\ to \reals, respectively. \dbstar\ denotes functions in the Darboux Baire*1 class; i.e., the class of all Darboux functions $f\RtoR$ with the property that for every perfect set $P$ there is its nonempty portion $Q=P\cap(a,b)$ such that $f$ restricted to $Q$ is continuous \cite{RO:Baire*1}. Any of the sets $\bigcup_{n\in\smallnats}(a_n,b_n)$ or $\bigcup_{n\in\smallnats}[a_n,b_n]$ is a {\em right interval set of a point\/} $a\in\reals$ if $a < b_{n+1} < a_n < b_n$ for $n\in\nats$ and $\lim_{n\to\infty}{a_n} = a$. In the case when $a=0$ we simply call it a {\em a right interval set}. Similarly we define a {\em left interval set} of a point $a\in\reals$. We say that $E$ is an {\em interval set\/} if it is the union of a right interval set and a left interval set. \section{\hskip -1em{.} Topologies} The ordinary topology on \reals\ is denoted by \ordinarytop. Next, the density topology, \idens\ topology and deep \idens\ topology are defined. For this, are needed the notions of a density point, \idens\ point and deep-\idens\ point. (See \cite{Oxtoby:MeasCat,Wilczynski:CatAn}. For a discussion of these ideas read \cite{CL:Refinements}.) Let $A\in\Lebesgue$. A number $x$, not necessarily in $A$, is a {\em density point\/} of $A$ if \begin{equation}\label{def_dens} \lim_{h\to 0^+}\frac{\measure{A\cap(x-h,x+h)}}{2h}=1. \end{equation} The set of all density points of $A\in\Lebesgue$ is denoted by \densitypts{A}. It is a consequence of the Lebesgue density theorem that for a measurable set $A$ almost every point of $A$ is a density point of $A$. Therefore, whenever $A$ is measurable, then so is \densitypts{A}\ and $\measure{A\triangle\densitypts{A}}=0$ \cite[p. 107]{WheedenZygmund:MeasureandIntegral}. It is also easy to see that \begin{equation}\label{con_dens} \densitypts{A}=\densitypts{B}\,\,\, \mbox{ for every $A,B\in\Lebesgue$ such that }\,\,\, A\triangle B\in\N. \end{equation} The family of sets \[ \densitytop=\{A\in\Lebesgue\colon A\subset\densitypts{A}\} \] forms a topology on \reals\ \cite{Oxtoby:MeasCat,GW:AppContTrans} called the {\em density topology\/}. If $A\in\BaireSets$, then $0$ is an {\em \idens\ point\/} of $A$ if for every increasing sequence $\{n_m\}_{m\in\smallnats}$ of natural numbers there exists a subsequence $\{n_{m_p}\}_{p\in\smallnats}$ such that \begin{equation}\label{def_idens1} \lim_{p\to\infty}\charf{n_{m_p}A\cap(-1,1)} = \charf{(-1,1)},\ \mbox{\I-a.e.} \end{equation} It is worth noticing that condition (\ref{def_idens1}) is equivalent to the fact that the set \begin{equation}\label{def_idens2} \liminf_{p\to\infty}n_{m_p}A = \bigcup_{q\in\smallnats} \bigcap_{p\ge q} n_{m_p}A \end{equation} is residual in $(-1,1)$. We say that a point $a$ is an \idens\ point of $A\in\BaireSets$ if $0$ is an \idens\ point of $A-a$. The set of all \idens\ points of $A\in\BaireSets$ is denoted by \idensitypts{A}. It is not difficult to see that $A\triangle\idensitypts{A}\in\I$ for every $A\in\BaireSets$ \cite[Theorem 3]{Wilczynski:CatAn} and that \begin{equation}\label{con_idens} \idensitypts{A}=\idensitypts{B}\,\,\, \mbox{ for every $A,B\in\BaireSets$ such that }\,\,\, A\triangle B\in\I. \end{equation} The family of sets \[ \itopology=\{A\in\BaireSets\colon A\subset\idensitypts{A}\} \] forms a topology on \reals\ \cite{PWW:CatAn,Wilczynski:CatAn} called the {\em \idens\ topology\/}. Finally, a point $a\in\reals$ is a {\em deep-\idens\ point\/} \cite{Wilczynski:CatAn} of an $A\in\BaireSets$ if there exists a closed set $F\subset A\cup \{a\}$ such that $a$ is an \idens\ point of $F$. The set of all deep-\idens\ points of $A\in\BaireSets$ is denoted by \deepidensitypts{A}. The family of sets \[ \deepitopology=\{A\in\BaireSets\colon A\subset\deepidensitypts{A}\} \] forms a topology on \reals\ called the {\em deep-\idens\ topology\/} \cite{Lazarow:Coarsest,Wilczynski:CatAn}. A point $x$ is a {\em dispersion} ({\em \I-dispersion}, {\em deep-\I-dispersion}) point of $A$ if $x$ is a density (\idens\, deep-\idens\ ) point of \complement{A}. In particular, $0$ is an \I-dispersion point of $A$ if for every increasing sequence $\{n_m\}_{m\in\smallnats}$ of natural numbers there exists a subsequence $\{n_{m_p}\}_{p\in\smallnats}$ such that \begin{equation}\label{con_idisp} (-1,1) \cap \bigcap_{q\in\smallnats} \bigcup_{p\ge q} n_{m_p}B = (-1,1) \cap \limsup_{p\to\infty}(n_{m_p}B) \in\I \end{equation} and $x$ is a deep-\I-dispersion point of $A$ if there exists an open $U\supset (A\setminus\{x\})$ such that $x$ is an \I-dispersion point of $U$. \bigskip We also need the following fact that easily follows from either of \cite[Lemma 2.4]{CL:Baire}, \cite[Theorem 1]{PWW:RemIDens} or \cite[Theorem 2]{Wilczynski:CatAn}. \proposition{sequence}{ If $E = \bigcup_{n\in\smallnats}[a_n,b_n]$ is a right interval set such that \begin{description} \item[(i)] $\lim_{n\to\infty}{(b_n - a_n)/a_n} = 0$ and \item[(ii)] $\lim_{n\to\infty}b_{n + 1}/a_n = 0,$ \end{description} then $0$ is a deep-\I-dispersion point of $E$. In particular, $E^c\in\deepitopology$. } The next theorem summarizes the topological properties of the topologies defined above. \theorem{topolprop}{The topologies \densitytop, \itopology\ and \deepitopology\ on \reals\ have the following properties: \begin{description} \item[(i)] $\ordinarytop\subset\densitytop$ and $\ordinarytop\subset\deepitopology\subset\itopology$; in particular, \densitytop, \itopology\ and \deepitopology\ are Hausdorff; \item[(ii)] a subset $C$ of \reals\ is closed and nowhere dense with respect to \densitytop\ if, and only if, $C\in\N$; a subset $C$ of \reals\ is closed and nowhere dense with respect to \itopology\ if, and only if, $C\in\I$; \item[(iii)] \densitytop\ and \deepitopology\ are completely regular; \itopology\ is not regular; \item[(iv)] \densitytop\ and \itopology\ are not separable; on the other hand, \deepitopology\ is separable; moreover, every set $D$ which is dense with respect to \ordinarytop\ is also dense with respect to \deepitopology; \item[(v)] every closed subinterval of \reals\ is connected in the topologies \densitytop, \itopology\ and \deepitopology; \item[(vi)] a set $A$ is compact with respect to any of the topologies \densitytop, \itopology\ and \deepitopology\ if, and only if, it is finite. \end{description} } Proof. (i) is obvious from the definitions. (ii) follows immediately from (\ref{con_dens}) and (\ref{con_idens}). The first part of (iii) can be found in \cite{Zachorski} or \cite{GNN:densitytop} for the density topology and in \cite{Lazarow:Coarsest} or \cite{Wilczynski:CatAn} for the deep-\idens\ topology. The second part follows from the fact that any set $D$ which is dense with respect to \ordinarytop\ cannot be separated in $\itopology$ from any point $x\in\reals\setminus D$ \cite[Theorem 5]{PWW:CatAn}. The above, together with the regularity of \deepitopology, also implies the second part of (iv), while its first part follows immediately from (ii). A proof of (v) can be found in \cite{GW:AppContTrans} in the case of the density topology and in \cite[Corollary 2]{PWW:CatAn} for the \idens\ topology. The deep-\idens\ topology case follows from that of \idens\ topology. To argue condition (vi) we need to show that an arbitrary infinite set is not compact with respect to any of \densitytop, \itopology\ or \deepitopology. So, let $A$ be an infinite set and let us choose a set $S=\{c_n\colon n\in\nats\}\subset A$ such that sequence $\{c_n\}_{n\in\smallnats}$ is monotone. By (ii), $S$ is closed with respect to \densitytop\ and \itopology. It is also closed in \deepitopology\, if $\{c_n\}_{n\in\smallnats}$ is unbounded. If $\{c_n\}_{n\in\smallnats}$ is bounded, say by $c$, and if we additionally assume that \[ \lim_{n\to\infty}\frac{c_{n+1}-c}{c_n-c} = 0, \] then, using Proposition \ref{sequence}, we can also show that $S$ is closed in \deepitopology. Now, the family of sets $W_m=\reals\setminus\{c_n\colon n\geq m\}\in\densitytop\cap\deepitopology$ forms an infinite open cover of $A$ without a finite subcover. \bigskip In the sequel the following two lemmas are needed. \lemma{lem:denseseq}{ There exists a decreasing sequence $S=\{s_n\}_{n\in\smallnats}$ converging to $0$ such that $0$ is not an \I-dispersion point of any set $B\in\itopology$ with the property that $B\cap[s_{n+1},s_n]\neq\emptyset$ for every $n\in\nats$. In particular, if $B\supset S$ then $0\not\in\idensitypts{\complement{B}}$. } Proof. Let \[ D_n=\{\frac{i}{2^n}\colon i=1,2,\ldots,2^n\}\subset (0,1] \] and let \[ S=\bigcup_{n\in\smallnats} \frac{1}{n} D_n. \] Notice that $S\subset (0,1]$ and that $S\setminus (0,\frac{1}{n}]$ is finite for every $n\in\nats$. Thus, S can be enumerated as a decreasing sequence converging to $0$. Now, let $B$ satisfy our assumptions. Then, $\tilde B$ also satisfies them and we may assume that $B=\tilde B$. Moreover, \[ nB\cap \left[\frac{i}{2^n},\frac{i+1}{2^n}\right]\neq\emptyset \] for every $i,n\in\nats$, $i<2^n$. In particular, for every increasing sequence $\{n_{m_p}\}_{p\in\smallnats}$ of positive integers, the set $\Cup_{p\in\smallnats} n_{m_p} B$ is dense and open. Hence, $\limsup_{p\in\smallnats} n_{m_p} B$ contains a dense \Gdelta\ subset of $(0,1)$. Thus, by (\ref{con_idisp}), $0$ cannot be an \idisp\ point of $B$. \lemma{lem:deepidisp}{ Let $P\subset(0,1]$ be closed and nowhere dense and let $\{b_n\}_{n\in\smallnats}$ be a sequence of positive numbers such that $\lim_{n\to\infty}b_{n + 1}/b_n = 0$. Then $0$ is a deep-\I-dispersion point of the set \[ Q=\bigcup_{n\in\smallnats}b_nP. \] In particular, $Q^c\in\deepitopology$. } Proof. According to Zaj\'{\i}\v{c}ek \cite[Theorem (5)]{Zajicek:AlternativeDefinition}, it is enough to prove that for every $c\in(0,1)$ there exist $\e>0$ and $\d>0$ such for any $x\in(0,\d)$ there exists a closed interval $I\subset\complement{Q}\cap (cx,x)$ such that $\measure{I}\geq x\e$. Let $c\in(0,1)$, $p=\min P$ and let $\d>0$ be such that \begin{equation}\label{eq:ddef} \frac{b_{n+1}}{b_n}0$ be a number such that for every interval $K\subset(0,1)$ of length $\geq p(1-c)/2$ there exists a closed interval $J\subset K\setminus P$ of length $\geq \e_0$. Such a number can be found, by partitioning $(0,1)$ into intervals $J_1,\ldots,J_k$, of length $< p(1-c)/4$ and defining \[ \e_0<\min\{\sup\{b-a\colon [a,b]\subset J_i\setminus P\}\colon 1\leq i\leq k\}. \] Put $\e=\min\{\e_0/2,(1-c)/2\}$. Now, let $x\in(0,\d)$ and define \[ m=\min\{n\colon pb_n\leq x\}. \] Then, by (\ref{eq:ddef}), $b_{m+1}a$ then $I=[xc,a]$ works, as $\measure{I}=x(1-c)/2\geq x\e$. Similarly, if $b_mp(1-c)/2$. Thus, by the definition of $\e_0$, there exists a closed interval $J\subset \frac{1}{b_m}(a,x)\setminus P$ of the length $\geq \e_0$. Hence, \[ I=b_mJ\subset(a,x)\setminus b_mP\subset(cx,x)\setminus b_m P =\complement{Q}\cap(cx,x) \] has length $\geq \e_0 b_m=2\e_0 b_m/2\geq x\e$. This finishes the proof. \bigskip This section is concluded with the fact that Theorem \ref{topolprop}(i) lists all possible inclusions between the topologies \ordinarytop, \densitytop, \itopology\ and \deepitopology. \theorem{th:densvsidens}{If ${\cal P}(\reals)$ stands for the discrete topology on \reals\ then \[ \begin{array}{ccccccc} \ordinarytop\cap\densitytop & \subset & \deepitopology\cap\densitytop & \subset & \itopology\cap\densitytop & \subset & \densitytop \nonumber \\ \| & & \cap & & \cap & & \cap \nonumber \\ \ordinarytop & \subset & \deepitopology & \subset & \itopology & \subset & {\cal P}(\reals) \nonumber \end{array} \] Moreover, all the inclusions are proper. } Proof. All the inclusions follow immediately from Theorem \ref{topolprop}(i). To show that the horizontal inclusions are proper, it is enough to argue for the inclusions in the first row. Thus, it is enough to prove that $\deepitopology\cap\densitytop\not\subset\ordinarytop$, $\itopology\cap\densitytop\not\subset\deepitopology$ and $\densitytop\not\subset\itopology$. To show that the vertical inclusions are proper we will show that $\deepitopology\not\subset\densitytop$. $\deepitopology\cap\densitytop\not\subset\ordinarytop$. Let $E = \bigcup_{n\in\smallnats}[a_n,b_n]$ be as in Proposition \ref{sequence}. It is easy to check that $0$ is also a dispersion point of $E$. So, $\complement{E}\in\deepitopology\cap\densitytop$ while, evidently, $\complement{E}\not\in\ordinarytop$. $\itopology\cap\densitytop\not\subset\deepitopology$. Let $E=\reals\setminus\rationals$. By Theorem \ref{topolprop}(ii) \rationals\ is closed in \itopology\ and \densitytop\ and so $E\in\itopology\cap\densitytop$. But, \rationals\ is also dense in \deepitopology. Hence, $E\not\in\deepitopology$. $\densitytop\not\subset\itopology$. Let $C$ be a nowhere dense set of positive Lebesgue measure. By the Lebesgue density theorem, $D=C\cap\densitypts{C}\in\densitytop$ and $D\neq\emptyset$. Moreover, by Theorem \ref{topolprop}(ii), $D$ is nowhere dense in \itopology, so $D\not\in\itopology$. $\deepitopology\not\subset\densitytop$. Let $C\subset [\frac{1}{2},1]$ be a closed nowhere dense set with positive Lebesgue measure and let $\{b_n\}_{n\in\smallnats}$ be a decreasing sequence of positive numbers such that $\lim_{n\to\infty}b_{n + 1}/b_n = 0$. Then, by Lemma \ref{lem:deepidisp}, $0$ is a deep-\I-dispersion point of $E=\bigcup_{n\in\smallnats}b_nC$ and $E^c\in\deepitopology$. On the other hand, for every $n\in\nats$, \[ \frac{m\left(E\cap (0,b_n)\right)}{b_n} = m\left(\inv{b_n} E\cap (0,1)\right) \geq m(C) >0. \] Thus, $0$ is not a dispersion point of $E$ and $E^c\not\in\densitytop$. \section{\hskip -1em{.} Classes of functions} For $\J,\K\in\{\N,\I,\D,\O\}$ let $C_{\cal JK}$ stand for the family of all continuous functions $f\colon (\reals,\T_{\cal J})\to(\reals,\T_{\cal K})$. It is easy to see that we have sixteen different classes: \[ \begin{array}{ccccccl} C_{\cal OO} & \,\,\, & C_{\cal DO} & & C_{\cal IO} & & C_{\cal NO} \nonumber \\ C_{\cal OD} & & C_{\cal DD} & \,\,\, & C_{\cal ID} & & C_{\cal ND} \nonumber \\ C_{\cal OI} & & C_{\cal DI} & & C_{\cal II} & \,\,\, & C_{\cal NI} \nonumber \\ C_{\cal ON} & & C_{\cal DN} & & C_{\cal IN} & & C_{\cal NN}. \nonumber \\ \end{array} \] Some of these classes have the following names associated with them: \begin{description} \item{$C_{\cal OO}=\cal C$} -- the class of (ordinary) continuous functions; \item{$C_{\cal NN}$} -- the class of density continuous functions; \item{$C_{\cal II}$} -- the class of \idens\ continuous functions; \item{$C_{\cal DD}$} -- the class of deep-\idens\ continuous functions; \item{$C_{\cal NO}$} -- the class of approximately continuous functions; \item{$C_{\cal IO}$} -- the class of \I-approximately continuous functions. \end{description} We start our investigation by showing that the other classes coincide with them or with the class of all constant functions. \theorem{const}{If \const\ stands for the class of all constant functions then \[ C_{\cal OD}=C_{\cal OI}=C_{\cal ON}=C_{\cal DI}=C_{\cal DN}=C_{\cal IN}= C_{\cal ND}=C_{\cal NI}=\const. \] } Proof. It is obvious that the constant functions are members of all these classes, as they are universally continuous. Thus, we need only show that the functions from the classes listed above $C_{\cal JK}$ are constant. By way of contradiction, let us assume that there is a nonconstant function $f$ in some class $C_{\cal JK}$ from the theorem. Case 1$^{\rm o}$. $C_{\cal JK}\in\{C_{\cal OD},C_{\cal OI},C_{\cal ON}\}$. By our assumption there are $a 0. \] Then $f$ is not deep-\idens\ continuous. } \example{cinfnotid}{There is an increasing homeomorphism $f\in\cinfinity\,\,\denscont\setminus\idenscont$.} Proof. Define \begin{equation}\label{def_convnotid} f(x)= \left\{ \begin{array}{ll} e^{-x^{-2}} & x>0 \\ 0 & x=0 \\ -e^{-x^{-2}} & x<0 \\ \end{array}\right. \end{equation} It is known that $f\in\cal C^{\infty}$ and that $f^{(n)}(0)=0$ for all $n\in\nats$. It is also easy to see that, for every $n\in\nats$, there exists $\varepsilon_n > 0$ such that $f^{(n)}(x)>0$ for every $x\in (0,\varepsilon_n)$. Thus, by Theorem \ref{th:Cinfinityex}, $f\not\in\idenscont$. It is also known \cite[Theorem 1]{CL:SpDensCont} that functions convex on open intervals are density continuous. This easily implies that $f$ is unilaterally density continuous at every point and so $f\in\denscont$. \bigskip Aversa and Wilczy\'nski \cite{AversaWilczynski:Homeomorphisms} study the homeomorphisms which preserve \idens\ points. In our terminology, the homeomorphism $h$ preserves \idens\ points when $\inv{h}\in\idenscont$. In particular, they proved the following theorem \cite[Corollary 1]{AversaWilczynski:Homeomorphisms}, which also follows easily from Theorem \ref{thm:genlipsch}. \theorem{thm:lipschitz}{If $h$ is a homeomorphism such that $h$ and $h^{-1}$ fulfill a local Lipschitz condition, then $h$ and $h^{-1}$ are \idens\/ continuous.} Now we are ready to present the next example. \example{notanal}{There is an increasing homeomorphism $h\in\idenscont\,\,\cinfinity\,\,\denscont\setminus\cal A$.} Proof. Let $h(x)=x+f(x)$, where $f$ is from (\ref{def_convnotid}). The function $f$ is \cinfinity\ with a pole, so $f\in\cinfinity\setminus\cal A$ and also $h\in\cinfinity\setminus\cal A$. But, evidently, $h$ is a homeomorphism such that $h$ and $h^{-1}$ fulfill a local Lipschitz condition. So, by Theorem \ref{thm:lipschitz}, $h\in\idenscont$ and, by its density analog \cite[Lemma 1]{CL:SpDensCont}, $h\in\denscont$. \bigskip For the last example of this section we need the following obvious fact. \proposition{pr:uniformconv}{ If $\{g_n\}$ is a sequence of \I-approximately (approximately) continuous functions converging uniformly to $g$ then the function $g$ is \I-approximately (approximately) continuous.} Proof. This is well known for approximately continuous functions \cite[Theorem 5.7]{Bruckner:DiffReal}. The same easy proof also works for the \I-approximately continuous case. \bigskip \example{notbstar}{There is a $g\in\iappr\,\,\appr\setminus\dbstar$.} Proof. Let $\{q_n\colon n\in\nats\}$ be an enumeration of \rationals. Let $f$ be as in (\ref{def_notcont}) and put \[ h_n(x)=\frac{2}{3^n}f(x-q_n). \] Then it is easy to see that $h_n(\reals)=[0,1/3^n]$ and that $h_n$ is piecewise linear at points $\neq q_n$ while $\osc(h_n,q_n)=1/3^n$. Define $g_n=\Sigma_{i\leq n}h_i$ and let $g$ be the limit of $\{g_n\}$. Notice that \[ |g-g_n|\leq \frac{1}{2}\frac{1}{3^n} \] so, $\{g_n\}$ converges to $g$ uniformly. The same argument as in Example \ref{notcont} shows that the functions $g_n\in\iappr\,\,\appr$. Thus, by Proposition \ref{pr:uniformconv}, $g\in\iappr\,\,\appr$. Also notice that, for every $n\in\nats$, \[ \osc(h,q_n)=\osc(h_n+(g-g_n),q_n) \geq\osc(h_n,q_n)-\sup_{x\in\smallreals}(g(x)-g_n(x)) \geq \frac{1}{3^n}-\frac{1}{3^{n+1}}>0 \] so, $g$ is discontinuous at every rational number. But, Baire*1 functions are continuous on a dense open set. Thus, $g\not\in\dbstar$. \section{\hskip -1em{.} \cinfinity\ deep-\idens\ continuous examples} We start this section with the following technical lemma. In what follows \nwdense\ stands for the ideal of nowhere dense subsets of \reals. \lemma{lem:longlist}{ Let $B\in\BaireSets$. The condition that $0$ is an \idisp\/ point of $B$ is equivalent to any of the following: \begin{description} \item[(i)] for every increasing sequence $\{n_k\}_{k\in\smallnats}$ of natural numbers there exists a subsequence $\{n_{k_p}\}_{p\in\smallnats}$ such that \[ \limsup_{p\to\infty}\left({n_{k_p}}\tilde B\right)\cap(-1,1)\in\I; \] \item[(ii)] for every increasing sequence $\{n_k\}_{k\in\smallnats}$ of natural numbers there exists a subsequence $\{n_{k_p}\}_{p\in\smallnats}$ such that \[ \limsup_{p\to\infty} \left(n_{k_p}\tilde{B}\right)\cap(-1,1)\in\nwdense; \] \item[(iii)] for every increasing sequence $\{n_k\}_{k\in\smallnats}$ of natural numbers and every nonempty interval $(a,b)\subset(-1,1)$ there exists a nonempty subinterval $(c,d)\subset(a,b)$ and a subsequence $\{n_{k_p}\}_{p\in\smallnats}$ such that for every $p\in\nats$ \[ (c,d)\cap n_{k_p}\tilde B = \emptyset; \] \item[(iv)] for every sequence $\{t_n\}_{n\in\smallnats}$ of positive numbers diverging to infinity and every nonempty interval $(a,b)\subset(-1,1)$ there exists a nonempty subinterval $(c,d)\subset(a,b)$ and a subsequence $\{t_{n_k}\}_{k\in\smallnats}$ such that for every $k\in\nats$ \[ (c,d)\cap t_{n_{k}}\tilde B = \emptyset; \] \item[(v)] for every sequence $\{t_n\}_{n\in\smallnats}$ of positive numbers diverging to infinity there exists a subsequence $\{t_{n_k}\}_{k\in\smallnats}$ such that for every nonempty interval $(a,b)\subset(-1,1)$ there exists a nonempty subinterval $(c,d)\subset(a,b)$ and $k_0\in\nats$ such that for all $k\ge k_0$ \[ (c,d)\cap t_{n_{k}}\tilde B = \emptyset. \] \end{description} } Proof. By (\ref{con_idens}) and (\ref{con_idisp}) condition (i) is equivalent to the fact that $0$ is an \idisp\ point of $B$. To finish the proof, it will be shown that (i) through (iv) each imply the next item and that (v) implies (i). (i)$\Rightarrow$(ii). Note that $\limsup_{p\to\infty}(n_{k_p}\tilde B)\cap(-1,1)$ is a \Gdelta\ set, and a \Gdelta\ set belongs to \I\ if, and only if, it belongs to \nwdense. (ii)$\Rightarrow$(iii). The sets \begin{equation}\label{decreasingsets} \{ \Cup_{p\ge m}n_{k_p}\tilde{B}: m\in\nats\} \end{equation} are open and their intersection is nowhere dense. If each of these open sets were dense in $(a,b)$, then their intersection would be a dense \Gdelta\ subset of $(a,b)$. Since this contradicts (ii), there must exist an $m_0\in\nats$ and $(c,d)\subset (a,b)$ such that $(c,d)\cap \Cup_{p\ge m_0}n_{k_p}\tilde{B}=\emptyset$. (iii) follows for the sequence $\{n_{k_p}\}_{p\geq m_0}$. (iii)$\Rightarrow$(iv). There is no generality lost with the assumptions that $t_1\ge 1$, that $t_{n+1}-t_n\ge 1$ for all $n$ and $(a,b)\subset [a,b]\subset (0,1)$. Let $p_n=\lfloor t_n\rfloor$, where $\lfloor x\rfloor$ stands for the greatest integer function. The sequence of integers $\{p_n\}$ satisfies the conditions of (iii), so there must exist a subsequence $\{p_{n_k}\}$ of $\{p_n\}$ and a nonempty $(c,d)\subset(a,b)$ such that $p_{n_k}\tilde{B}\cap(c,d)=\emptyset$ for all $k\in\nats$. Since $t_{n_k}/p_{n_k}\to 1$ as $k\to\infty$, there exists a $k_0$ such that \[ 1\le\frac{t_{n_k}}{p_{n_k}}<1+\frac{d-c}{3c}, \ \ \mbox{ for all } \ \ k\ge k_0. \] Let $J=(c+(d-c)/3,d)$ and $k\ge k_0$. Then \[ \emptyset = \frac{t_{n_k}}{p_{n_k}}\left( (c,d)\cap p_{n_k}\tilde{B}\right) = \left(\frac{t_{n_k}c}{p_{n_k}},\frac{t_{n_k}d}{p_{n_k}}\right) \cap t_{n_k}\tilde{B} \supset J\cap t_{n_k}\tilde{B}. \] Part (iv) follows easily from this. (iv)$\Rightarrow$(v). Without loss of generality we may assume that $a$ and $b$ are rational. Let $\{(a_s,b_s)\}_{s\in\smallnats}$ be an enumeration of all such intervals, and let us choose a sequence $\{t_n\}$. The idea of the proof is an application of (iv) infinitely many times, and diagonalization. Let $m_k^0 = k$ for $k\in\nats$. We will, by induction on $s\in\nats$, construct sequences $\{m_k^s\}_{k\in\smallnats}$ such that, for every $s\in\nats$, $\{m_k^s\}_{k\in\smallnats}$ will be a subsequence of $\{m_k^{s-1}\}_{k\in\smallnats}$, and there will be a nonempty interval $(c,d)$ (possibly dependent on $s$) contained in $(a_s,b_s)$ such that for all $k\in\nats$ \[ (c,d)\cap(t_{m_k^s}\tilde B) = \emptyset. \] The construction is facilitated by (iv). Now, define $t_{n_k} = t_{m_k^k}$. Then, for every $k_0\in\nats$, there exists $(c,d)\subset(a_{k_0},b_{k_0})$ such that for every $k \geq k_0$ \[ (c,d)\cap(t_{n_k}\tilde B) = \emptyset. \] This proves (v). (v)$\Rightarrow$(i). Let $\{n_k\}_{k\in\smallnats}$ be an increasing sequence of natural numbers and let $\{(a_p,b_p)\}_{p\in\smallnats}$ be an enumeration of all nonempty subintervals of $(-1,1)$ with rational endpoints. Using (v), there is a subsequence $\{n_k^1\}$ of $\{n_k\}$ and a nonempty open interval $U_1\subset (a_1,b_1)$ such that \[ U_1\cap n_k^1\tilde B=\emptyset,\ \ \mbox{ for all } \ \ k\in\nats. \] Continuing inductively, for each $p>1$ we can choose a subsequence $\{n_k^p\}$ of $\{n_k^{p-1}\}$ and a nonempty interval $U_p\subset (a_p,b_p)$ such that \begin{equation}\label{eq:emptyIntersection} U_p\cap n_k^p\tilde B=\emptyset,\ \ \mbox{ for all } \ \ k\in\nats. \end{equation} If $n_{k_p}=n_p^p$ then (\ref{eq:emptyIntersection}) implies \[ \Cup_{k\in\smallnats}U_k\cap\limsup_{p\to\infty} n_{k_p}\tilde{B}=\emptyset. \] Since $\Cup_{k\in\smallnats}U_k$ is a dense open subset of $(-1,1)$, this implies (i). This finishes the proof of Lemma \ref{lem:longlist}. \bigskip Notice that Lemma \ref{lem:longlist} can be viewed as a version of \cite[Lemma 1]{PWW:RemIDens}. (Compare also \cite[Corollary 1]{PWW:RemIDens}.) In particular, an alternative prove of Lemma \ref{lem:longlist} is as follows. Implication ``$0$ is an \I-dispersion point of $B$''$\Rightarrow$(v) is implicitely contained in the proof of \cite[Lemma 1]{PWW:RemIDens}. Since the implications (v)$\Rightarrow$(iv)$\Rightarrow$(iii) and (ii)$\Rightarrow$(i)$\Rightarrow$``$0$ is an \I-dispersion point of $B$'' are obvious, it is enough to show (iii)$\Rightarrow$(ii). But it is included in this paper as a proof of (v)$\Rightarrow$(i). \bigskip In what follows we also need the following theorem. Notice that Theorem \ref{thm:lipschitz} that can be obtained from the following theorem by taking an arbitrary set $E$ and $u_k=1$. \theorem{thm:genlipsch}{Let $0$ be a right \idens\ point of a right interval set $E=\Cup_{k\in\smallnats} (a_k,b_k)\subset[0,1]$ and let $h\colon[-1,1]\to[-1,1]$ be such that the restricted function $h|_{(a_k,b_k)}$ is a homeomorphism for every $k\in\nats$. Moreover, let us assume that there % I still think, it should be plural exist a nondecreasing sequence $\{u_k\}_{k\in\smallnats}$ of positive numbers and constants $K,L>0$ such that for every $k\in\nats$ the functions \[ h_k=u_k h|_{\{0\}\cup(a_k,b_k)}\colon \{0\}\cup(a_k,b_k)\to [-K,K] \quad\mbox{and}\quad \left(h_k|_{(a_k,b_k)}\right)^{-1} \] satisfy a Lipschitz condition with constant $L$. Then $h$ is right \idens\ continuous at $0$. } Before proving the theorem let us notice that it implies the following corollary. It can be considered as an \idens\ analog of the theorem that convex functions defined on open intervals are density continuous \cite[Theorem 1]{CL:SpDensCont}. Notice also that an easy modification of the Example \ref{cinfnotid} shows that not all convex functions defined on open intervals are \idens\ continuous. \corollary{cor:genlinear}{Let $0$ be a right \idens\ point of a right interval set $\Cup_{k\in\smallnats} (a_k,b_k)$ and let $h\colon[-1,1]\to[-1,1]$ be a convex function such that the restricted functions $h|_{(a_k,b_k)}$ are linear for every $k\in\nats$. Then $h$ is right \idens\ continuous at $0$. } Proof. Let $c_k=h^{\prime}((a_k+b_k)/2)$ for $k\in\nats$. Then, by the convexity of $h$, the sequence $\{c_k\}$ is nonincreasing. Let, $c=\inf_{k\in\smallnats}c_k$. Notice that $c\neq -\infty$, as $c=-\infty$ would contradict the convexity of $h$ on $(-1,1)$. If $c\neq 0$ then $h$ and $\inv{h}$ both satisfy a Lipschitz condition on a right neighbourhood of $0$. By Theorem \ref{thm:lipschitz}, this implies that $h$ is \idens\ continuous at $0$. (See the remark before Theorem \ref{thm:genlipsch}.) If $c=0$, we may also assume that $h(0)=0$. There are two cases to consider. If $P=\{k:c_k=0\}$ is infinite, then $h$ must be linear on a right-hand neighborhood of $0$ since $h$ is convex on $(0,1)$. Otherwise, assume $P$ is finite. In this case there is no generality lost in assuming that $P=\emptyset$. Evidently, the sequence $u_k=c_k^{-1}$ is nondecreasing and the derivative of $u_kh$ at points from $(a_k,b_k)$ equals $1$. Moreover, the convexity of $h$ implies that $h_k=u_k h|_{\{0\}\cup(a_k,b_k)}\colon \{0\}\cup(a_k,b_k)\to [0,1]$. An application of Theorem \ref{thm:genlipsch} finishes the proof. \bigskip We also need the following easy lemma. \lemma{nest}{ Let $\varepsilon, M >0$ and let $\{(a_n,b_n)\}_{n\in\nats}$ be a sequence of subintervals of $[-M,M]$ such that $b_n-a_n>\varepsilon$ for every $n\in\nats$. Then there exists a nonempty interval $(a,b)\subset [-M,M]$ and an increasing sequence $\{n_k\}_{k\in\nats}$ of positive numbers such that $(a,b)\subset (a_{n_k},b_{n_k})$ for every $k\in\nats$. } Proof. Let $c_n$ be the center of $(a_n,b_n)$ for every $n\in\nats$ and let $-M=d_00$ for $x\in D$. To choose such a function it is enough to define $f$ on $(l_n,r_n)$ by \[ f(x)=u_n e^{-(x-l_n)^{-2}-(x-r_n)^{-2}}, \] which is known to be \cinfinity\ with poles at $l_n$ and $r_n$, and in which the constants $u_n\leq c_n$ are chosen in such a way that $f^{(i)}(x)\leq\frac{1}{n}$ for every $i\leq n$ and $x\in[l_n,r_n]$. Let $g(x)=\int_0^x f(y)\,dy$ and $h(x)=\int_0^x g(y)\,dy$. Evidently $h$ is a \cinfinity\ increasing and convex homeomorphism satisfying (1). It is also easy to see that (2) holds, as $f(x)\leq u_n\leq c_n$ for all $x\in[0,b_n]\subset[0,1]$. Moreover, by Theorem \ref{thm:lipschitz}, $h$ is \idens\ continuous at every point $\neq 0$ and, by Corollary \ref{cor:genlinear}, $h$ is right \idens\ continuous at $0$. The function $h$ is density continuous, since it can be extended to a convex function on an open interval \cite[Theorem 1]{CL:SpDensCont}. \example{ex:filings}{Let $k0$ on the modification set. Thus, $g$ is \idens\ continuous. \example{deepInotI}{There is an $f\in\deepidenscont\,\,\cinfinity\,\,\denscont\setminus\idenscont$.} Proof. Let $g\colon[0,1]\to[0,1]$ be as in Example \ref{ex:filings}, constant on an interval containing $1/2$ and choose constants $c_k\leq\frac{1}{k}$ such that $c_kg^{(i)}(x)\leq\frac{1}{k}$ for every $x\in [0,1]$ and $i\leq k$. Next, choose a right interval set $E=\Cup_{k\in\smallnats} [p_k,q_k]\subset[0,1]$ for which $0$ is a right \idens\ point and let $h$ and $\{d_k\}$ be choosen as in Example \ref{ex:idensewithpole} for the sequence $\{c_k\}$ and the set $E$. Extend $h$ onto \reals\ by putting $h(x)=0$ for $x<0$ and as a linear function on $[q_1,\infty)$ by the same formula as on $[p_1,q_1]$. It is easy to see that \[ h\in\deepidenscont\,\,\cinfinity\,\,\denscont. \] Notice that for every $t>0$ the function $g_{k,t}\colon [0,t]\to[0,t]$ defined by $g_{k,t}(x)=t\,d_kg(x/t)$ has the properties that \[ g_{k,t}^{(i)}(x)\leq\frac{1}{k}\mbox{ for every }x\in [0,t] \mbox{ and }i\leq k \] and \[ g_{k,t}(x)=d_k x\mbox{ for every }x\in t\left( [0,l]\cup[b,1]\right). \] Moreover, let $S=\{s_n\}_{n\in\nats}$ be from Lemma \ref{lem:denseseq}. Without loss of generality we may assume that $S\subset\interior{E}$, as $0$ is a right deep-\idens\ point of $E$. Notice, that each component of $E$ contains only finitely many points from $S$. Denote the component of $E$ which contains $s_n$ by $[p_{k(n)},q_{k(n)}]$. We need also to choose nonempty pairwise disjoint intervals $(a_n,b_n)\subset [a_n,b_n]\subset(p_{k(n)},q_{k(n)})$, centered at $s_n$ such that $0$ is a dispersion point of $P=\UoverN(a_n,b_n)$ and \begin{equation}\label{condsmall} \lim_{n\to\infty} \frac{h(b_n)-h(a_n)}{h(a_n)}=0. \end{equation} Define $h_n(x)=g_{k(n),b_n-a_n}(x-a_n)$ for $x\in(a_n,b_n)$, with the interval in $(a_n,b_n)$ on which $h_n$ is constant denoted by $I_n$. Notice that $s_n\in I_n$. Define $f$ by putting $f|_{P^c}=h|_{P^c}$ and $f(x)=h_n(x) + h(a_n)$ for $x\in(a_n,b_n)$. Obviously, $f$ is \cinfinity, \idens\ continuous and density continuous at any point $\neq 0$. It is also clear, by the choice of the constants $c_k$, that $f$ is infinitely differentiable at $0$. To see that $f$ is density continuous at $0$ it is enough to notice that $f|_{(-\infty,0]\cup\interior{E}}=h|_{(-\infty,0]\cup\interior{E}}$, while $h\in\denscont$ and $(-\infty,0]\cup\interior{E}\in\densitytop$. It is also easy to see that $f\not\in\idenscont$, as $0$ is an \I-dispersion point of the countable set $f(S)$, while $0$ is not an \I-dispersion point of $f^{-1}(f(S))\supset \Cup_{n\in\nats}I_n\supset S$. It remains to prove that $f$ is right deep-\idens\ continuous at $0$, as the left hand case is obvious. So, let $0$ be an \I-dispersion point of a right interval set $B=\Cup_{n\in\nats} (v_n,w_n)$. It is enough to prove that $0$ is an \I-dispersion point of $\inv{f}(B)$, as the sets $E\cup\{0\}\in\itopology$, where $E$ is an open interval set, form a basis of \deepitopology\ at $0$ \cite{Lazarow:Coarsest}. We claim that the lengths of some of the intervals $(v_n,w_n)$ can be increased slightly to satisfy the condition \begin{equation}\label{con:inclus} (h(a_k),h(b_k))\cap(v_n,w_n)=\emptyset\,\, \mbox{ or }\,\, (h(a_k),h(b_k))\subset(v_n,w_n), \end{equation} while still preserving the property that $0$ is an \I-dispersion point of $B$. To see this, define $B^\prime= B\cup\Cup\{(h(a_k),h(b_k))\colon (h(a_k),h(b_k))\cap B\neq\emptyset\}$. Evidently, $B^\prime$ satisfies (\ref{con:inclus}). We have to prove only that $0$ is an \I-dispersion point of $B^\prime$. So, let $(a,b)\subset(0,1)$ and $\{n_k\}_{k\in\smallnats}$ be an increasing sequence of natural numbers. Using Lemma \ref{lem:longlist}(iii), we can find an increasing subsequence $\{n_{k_p}\}_{p\in\smallnats}$ and a nonempty interval $(c,d)\subset(a,b)$ such that \[ (c,d)\cap n_{k_p} B=\emptyset\,\,\mbox{ for all }\,\,p\in\nats. \] According to (\ref{condsmall}), there is an $p_0\in\nats$ such that \[ \frac {h(b_{n_{k_p}})-h(a_{n_{k_p}})} {h(a_{n_{k_p}})}<\frac{d-c}{3} \,\,\,\mbox{ for all }\,\,\, p\geq p_0. \] But, then it is clear that \[ (c+\frac{d-c}{3},d-\frac{d-c}{3})\cap n_{k_p} B^\prime= \emptyset\,\,\mbox{ for all }\,\,p\geq p_0, \] so, by Lemma \ref{lem:longlist}(iii), $0$ is an \I-dispersion point of $B^\prime$. But (\ref{con:inclus}) implies that $h^{-1}(B)=f^{-1}(B)$, so $0$ is a deep-\I-dispersion point of $f^{-1}(B)$, as $h$ is deep-\idens\ continuous. \example{cinfnotdendcont}{There is an $f\in\idenscont\,\,\cinfinity\setminus\denscont$.} Proof. We will proceed in a method similar to Example \ref{deepInotI}. Let $g\colon[0,1]\to[0,1]$ be as in Example \ref{ex:filings} such that $g$ is constant on $[1/3,2/3]$. Choose constants $c_k\leq\frac{1}{k}$ such that $c_kg^{(i)}(x)\leq\frac{1}{k}$ for every $x\in [0,1]$ and $i\leq k$. Next, choose a right interval set $E=\Cup_{k\in\smallnats} [p_k,q_k]\subset[0,1]$ for which $0$ is a right density and deep-\idens\ point. Let $h$ and $\{d_k\}$ be choosen as in Example \ref{ex:idensewithpole} for the sequence $\{c_k\}$ and set $E$. Extend $h$ onto \reals\ by putting $h(x)=0$ for $x<0$ and as a linear function on $[q_1,\infty)$ by the same formula as on $[p_1,q_1]$. It is easy to see that \[ h\in\idenscont\,\,\cinfinity\,\,\denscont. \] Notice that for every $t>0$ the function $g_{k,t}\colon [0,t]\to[0,t]$ defined by $g_{k,t}(x)=t\,d_kg(x/t)$ has the properties that \begin{equation}\label{con:gfunct1} g_{k,t}^{(i)}(x)\leq\frac{1}{k}\mbox{ for every }x\in [0,t] \mbox{ and }i\leq k \end{equation} and \begin{equation}\label{con:gfunct2} g_{k,t}(x)=d_k x\mbox{ for every }x\in t\left( [0,l]\cup[b,1]\right). \end{equation} Moreover, let $D=\UoverN [a_n,b_n]$ be a right interval set such that $D^c\in\deepitopology\setminus\densitytop$. (See Theorem \ref{th:densvsidens}.) Without loss of generality we may assume that $D\subset\interior{E}$, as $0$ is a right density and deep-\idens\ point of $E$. Notice, that each component of $E$ contains only finitely many intervals $[a_n,b_n]$. Denote the component of $E$ which contains $[a_n,b_n]$ by $[p_{k(n)},q_{k(n)}]$. Let $h_n=g_{k(n),b_n-a_n}$, with the interval in $(0,b_n-a_n)$ on which $h_n$ is constant denoted by $I_n$. Define $f$ by putting $f|_{\left(\UoverN(a_n,b_n)\right)^c}=h|_{\left(\UoverN(a_n,b_n)\right)^c}$ and $f(x)=h_n(x-a_n) + h(a_n)$ for $x\in[a_n,b_n]$. Obviously $f$ is \cinfinity, \idens\ continuous and density continuous at any point $\neq 0$. It is also clear, by the choice of the constants $c_k$, that $f$ is infinitely differentiable at $0$. Also, $f$ is \idens\ continuous at $0$ as $f|_{D^c}=h|_{D^c}$, while $h\in\idenscont$ and $D^c\in\itopology$. Finally, $f$ is not density continuous at $0$, as $0$ is a dispersion point of the countable set $f(\UoverN I_n)$, while $0$ is not a dispersion point of a set $f^{-1}(f(\UoverN I_n))=\UoverN I_n$. This last statement is correct, as $\UoverN I_n$ is obtained by taking the middle third from every component of $D=\UoverN [a_n,b_n]$. \bigskip \bibliographystyle{plain} %\bibliography{examples} \begin{thebibliography}{10} \bibitem{AversaWilczynski:Homeomorphisms} V.~Aversa and W.~Wilczy\'nski. \newblock Homeomorphisms preserving {$\cal {I}$}-density points. \newblock {\em Boll. Un. Mat. Ital.}, B(7)1:275--285, 1987. \bibitem{Bruckner:DiffReal} A.~M. Bruckner. \newblock {\em Differentiation of Real Functions}. \newblock Lecture Notes in Mathematics 659. Springer-Verlag, 1978. \bibitem{KC:Cndfunctions} Krzysztof Ciesielski. \newblock Density-to-deep-{$\cal {I}$}-density continuous functions. \newblock {\em Real Anal. Exch.} \newblock this volume. \bibitem{CL:Analytic} Krzysztof Ciesielski and Lee Larson. \newblock Analytic functions are {$\cal {I}$}-density continuous. \newblock (submitted). \bibitem{CL:Baire} Krzysztof Ciesielski and Lee Larson. \newblock Baire classification of {$\cal {I}$}-approximately and {$\cal {I}$}-density continuous functions. \newblock (submitted). \bibitem{CL:Refinements} Krzysztof Ciesielski and Lee Larson. \newblock Refinements of the density and {$\cal {I}$}-density topologies. \newblock (submitted). \bibitem{CL:SpDensCont} Krzysztof Ciesielski and Lee Larson. \newblock The space of density continuous functions. \newblock {\em Acta Math. Hung.}, to appear. \bibitem{CLO:DensContCont} Krzysztof Ciesielski, Lee Larson, and Krzysztof Ostaszewski. \newblock Density continuity versus continuity. \newblock {\em Forum Mathematicum}, 2:265--275, 1990. \bibitem{GNN:densitytop} C.~Goffman, C.~J. Neugebauer, and T.~Nishiura. \newblock The density topology and approximate continuity. \newblock {\em Duke Math. J.}, 28:497--506, 1961. \bibitem{GW:AppContTrans} Casper Goffman and Daniel Waterman. \newblock Approximately continuous transformations. \newblock {\em Proc. Amer. Math. Soc.}, 12:116--121, 1961. \bibitem{Lazarow:Coarsest} E.~{\L}azarow. \newblock The coarsest topology for {$\cal {I}$}-approximately continuous functions. \newblock {\em Comment. Math. Univ. Caroli.}, 27(4):695--704, 1986. \bibitem{RO:Baire*1} R.~J. O'Malley. \newblock {B}aire*1 {D}arboux functions. \newblock {\em Proc. Amer. Math. Soc.}, 60:187--192, 1976. \bibitem{Oxtoby:MeasCat} J.~C. Oxtoby. \newblock {\em Measure and Category}. \newblock Springer-Verlag, 1971. \bibitem{PWW:CatAn} W.~Poreda, E.~Wagner-Bojakowska, and W.~Wilczy\'{n}ski. \newblock A category analogue of the density topology. \newblock {\em Fund. Math.}, 75:167--173, 1985. \bibitem{PWW:RemIDens} W.~Poreda, E.~Wagner-Bojakowska, and W.~Wilczy\'{n}ski. \newblock Remarks on {$\cal {I}$}-density and {$\cal {I}$}-approximately continuous functions. \newblock {\em Comm. Math. Univ. Carolinae}, 26(3):553--563, 1985. \bibitem{WheedenZygmund:MeasureandIntegral} Richard~L. Wheeden and Antoni Zygmund. \newblock {\em Measure and Integral}. \newblock Marcel Dekker, Inc., 1977. \bibitem{Wilczynski:CatAn} W.~Wilczy{\'n}ski. \newblock A category analogue of the density topology, approximate continuity, and the approximate derivative. \newblock {\em Real Anal. Exch.}, 10:241--265, 1984-85. \bibitem{Willard:GenTop} S.~Willard. \newblock {\em General Topology}. \newblock Springer, 1971. \bibitem{Zachorski} Z.~Zachorski. \newblock Sur la premi\`ere d\'eriv\'ee. \newblock {\em Trans. Amer. Math. Soc.}, 69:1--54, 1950. \bibitem{Zajicek:AlternativeDefinition} L.~Zaj\'{\i}\v{c}ek. \newblock Alternative definitions of the {$J$}-density topology. \newblock {\em Acta Univ. Carolinae--Mat. et Phys.}, 28(1):57--61, 1987. \end{thebibliography} \end{document}