% Category theorems concerning I-density continuous functions.
\documentstyle[12pt]{article} \pagestyle{myheadings}

\input{epsf.tex}

\newcommand{\putepsf}[5]{
     \begin{figure}[htb]
     \[
         \vbox to #2in{\hrule width #1in height 0pt depth 0pt \vfill
         \special{illustration #3 scaled 250}}
     \]
     \caption{#4\label{#5}}
     \end{figure}
}




%\input{imacros} 
%%%%%%%%%%%%%%%%%%%%%   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}
% natural numbers
     \newcommand{\nats}{\mbox{{\amssymten N}}}
     \newcommand{\smallnats}{\mbox{{\amssymnine N}}}
% rational numbers
     \newcommand{\rationals}{\mbox{{\amssymten Q}}}
     \newcommand{\smallrationals}{\mbox{{\amssymnine Q}}}
% real numbers
     \newcommand{\reals}{\mbox{{\amssymten R}}}
     \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{$\cal J$}}
%nowhere dense sets
     \newcommand{\nwdense}{\mbox{$\I_0$}}
%sets with cardinality < continuum
     \newcommand{\lessthanc}{\mbox{${\cal I}_c$}}
%empty ideal
     \newcommand{\emptyideal}{\mbox{$\cal 0$}}
 
%%%%%%%%%%%%%%%%%%%   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{$\T_{\cal I}$}}
%deep I-density topology
     \newcommand{\deepitopology}{\mbox{$\T_{\cal D}$}}
 
%%%%%%%%%%%%%%%%%%%%%  FUNCTION SPACES
 
     \newcommand{\C}{\mbox{$\cal C$}}
%continuous functions
     \newcommand{\ocont}{\mbox{$\C_{\cal OO}$}}
%approximately continuous functions
     \newcommand{\acont}{\mbox{$\C_{\cal NO}$}}
%density continuous functions
     \newcommand{\dcont}{\mbox{$\C_{\cal NN}$}}
%I-dens cont funct
     \newcommand{\idenscont}{\mbox{$\C_{\cal II}$}}
%deep I-dens cont funct
     \newcommand{\deepidenscont}{\mbox{$\C_{\cal DD}$}}
% Baire star functions
     \newcommand{\bstar}{\mbox{Baire*1}}
% Infinitely differentiable functions
     \newcommand{\cinfinity}{\mbox{$\C^\infty$}}
 
%%%%%%%%%%%%%%%%%%%%%% 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{$#2\circ #1\circ\inv{#2}$}}
%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{$\mbox{m}(#1)$}}
 
%%%%%%%%%%%%%%%%%%%%%% THEOREM-LIKE THINGS
 
%Note the capitals to differentiate them from the commands.
 
\newtheorem{Theorem}{Theorem}
\newtheorem{Observation}[Theorem]{Observation}
\newtheorem{Lemma}[Theorem]{Lemma}
\newtheorem{Corollary}[Theorem]{Corollary}
\newtheorem{Proposition}[Theorem]{Proposition}
\newtheorem{Example}[Theorem]{Example}
\newtheorem{Problem}[Theorem]{Problem}
 
%define shorter commands for these environments
 
\newcommand{\theorem}[2]{\begin{Theorem}\Label{#1}#2\end{Theorem}}
\newcommand{\lemma}[2]{\begin{Lemma}\Label{#1}#2\end{Lemma}}
\newcommand{\corollary}[2]{\begin{Corollary}\Label{#1}#2\end{Corollary}}
\newcommand{\example}[2]{\begin{Example}\Label{#1}#2\end{Example}}
\newcommand{\proposition}[2]{\begin{Proposition}\Label{#1}#2\end{Proposition}}
\newcommand{\observation}[2]{\begin{Observation}\Label{#1}#2\end{Observation}}
\newcommand{\problem}[2]{\begin{Problem}\Label{#1}#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-den\-sity}
\newcommand{\idisp}{\I-dispersion}
 
\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{\Fsigma}{\mbox{$\mbox{\bf F}_\sigma$}}
 
\newcommand{\Y}{\mbox{$\Psi$}}

\newcommand{\Lazarow}{{\L}azarow}

%measurable functions
     \newcommand{\measureF}{\mbox{${\cal F}_m$}}
%Baire functions
     \newcommand{\BaireF}{\mbox{${\cal F}_c$}}
     
     \newcommand{\calS}{\mbox{$\cal S$}}

%%%%%%%%%%%%%%%%%%%%%%% Change markers

\newcommand{\change}{\marginpar{!!!CHANGE}}
\newcommand{\changestart}{\marginpar{!!!BEGIN}}
\newcommand{\changeend}{\marginpar{!!!END}}
\newcommand{\changefigure}{\marginpar{PUT PICTURE}}




\markright{Category Theorems}


\title{Category Theorems Concerning \I-density Continuous Functions} 
\author{}
\date{}
 
\begin{document}
\maketitle
 
\begin{flushleft}
{\small
\noindent Krzysztof Ciesielski,
  Department of Mathematics, West Virginia University, Morgantown, WV 26506
%      \footnote{ This author was partially supported by
%                West Virginia University Senate Research Grant.}

\noindent Lee Larson,
Department of Mathematics, University of Louisville, Louisville, KY 40292
    \footnote{
     This author was partially supported by a University of Louisville
     Arts and Sciences Research grant.
     \par\vskip 3pt
     \begin{flushleft}
          Key Words: \idens\ topology, \idens\ continuous functions,
                     first category sets.\\
          AMS Subject Classification. Primary: 54H15; Secondary 20M20, 26A18.
     \end{flushleft}
    }
 
}
\end{flushleft}
 
\begin{abstract}
The \I-density topology \itopology\ on \reals\ is a refinement of the 
natural topology. It is a category analogue of the density topology
\cite{PWW:CatAn,Wilczynski:CatAn}.
This paper is concerned with \I-density continuous functions; i.e.,
the real functions that are continuous when the \I-density topology is used 
on the domain and the range. It is shown, that the family
${\cal C}_{\cal I}$ of ordinary continuous functions $f\colon [0,1]\to\reals$
which have at least one point of \I-density continuity is a first category subset of
${\cal C}([0,1])=\{f\colon [0,1]\to\reals\colon f\mbox{ is continuous}\}$
equipped with the uniform norm. It is also proved, that the class
\idenscont\ of \I-density continuous functions, equipped with the topology 
of uniform convergence, is of first category in itself. These results 
remain true when the \I-density topology is replaced by the 
deep \I-density topology.
\end{abstract}

\section{Introduction}

The ordinary density topology on \reals\ is defined to be the collection of all
subsets of \reals\ which have full Lebesgue density at every point
\cite{Bruckner:DiffReal}. The collection of all sets open in the density topology
is written as \densitytop.
The open sets in the ordinary topology are written \ordinarytop.
A function $f\RtoR$ is {\em approximately continuous} at a point $x$, if it is
continuous at $x$ with the ordinary topology on the range and the density
topology on the domain and it is {\em density continuous} at the point $x$, if it
is continuous at $x$ when \densitytop\ is used on both the domain and the range.
The spaces of everywhere ordinary continuous, approximately continuous and
density continuous functions $f\RtoR$ are written as \ocont, \acont\ and \dcont,
respectively.

The structure of \ocont\ and \acont\ are quite well understood, but \dcont\ is
more difficult to study, mainly because it is closed neither under addition nor
uniform convergence \cite{CL:SpDensCont}. In particular, the relationship
between density continuity and ordinary continuity is quite complicated. The
definitions yield at once that $\ocont\subset\acont\supset\dcont$, 
but it is not hard to construct examples showing that
\begin{equation}\label{eq:nocontainments}
\ocont\not\subset\dcont\not\subset\ocont
\end{equation}
\cite{CL:SpDensCont,CL:Examples}. The following
theorem is known \cite{CLO:DensContCont}.

\theorem{thm:dfirstcat}{Let \ocont\ be given the topology of uniform
convergence. If $\cal C$ is the subset of \ocont\ consisting of functions with
at least one point of density continuity, then $\cal C$ is a first category
subset of \ocont.}

A combination of this theorem with the fact that every density continuous
function is continuous on a dense open set can be used to show the following
corollary \cite{CLO:DensContCont}.

\corollary{cor:dinself}{If \dcont\ is given the topology of uniform
convergence, then it is a first category subset of itself.}

Let \I\ be the collection of all first category subsets of \reals\ and
$E\subset\reals$. A point $x\in\reals$ is an {\em \idisp\ point} of $E$ if, and
only if, for every increasing sequence of natural numbers $\{ t_n\}$ there is a
subsequence $\{ t_{n_m}\}$ such that
\[
\limsup_{m\in\smallnats}t_{n_m}(E-x)\cap(-1,1)\in\I.
\]
The point $x$ is an \idens\ point of $E$ if, and only if, it is an \idisp\
point of \complement{E}. Using this category density instead of Lebesgue
density, the {\em\idens\ topology}, \itopology\ is defined to consist of all
Baire sets $E\subset\reals$ such that every point of $E$ is an \idens\ point of
$E$ \cite{PWW:CatAn,Wilczynski:CatAn}. 

\itopology\ has many properties in common
with \densitytop, but \densitytop\ is completely regular while \itopology\ is
not. To remedy this, a topology 
coarser than \itopology, called the {\em deep
\idens\ topology} is introduced in the following way. A point $x$ is a {\em deep
\idens\ point} of the set $E\subset\reals$ if, and only if, there is an
ordinary closed set $F\subset E\cup\{x\}$ such that $x$ is an \idens\ point of
$F$. Using the idea of deep \idens, the {\em deep \idens\ topology,}
\deepitopology, is defined in the by now familiar way \cite{Lazarow:Coarsest}.
\deepitopology\ is completely regular \cite{Lazarow:Coarsest}.

Given these two topologies based on \idens, the \idens\ continuous functions,
\idenscont, and deep \idens\ continuous functions, \deepidenscont, are defined
in the natural way.

It is reasonable to ask if the known properties of the density continuous
functions can be proved in the case of the \idens\ and deep \idens\ continuous
functions. The purpose of this paper is to establish Theorem \ref{thm:dfirstcat}
and Corollary \ref{cor:dinself} using these topologies in place of the density
topology.

\section{Comparison with \ocont}

The purpose of this section is to prove that  the \idens\ continuous and deep
\idens\ continuous functions have the same relationship to the ordinary
continuous functions as do the density continuous functions. First, in analogy
to (\ref{eq:nocontainments}), it is known that the following
relationships hold \cite{CL:Examples}.
\begin{equation}\label{eq:inocontainments}
\idenscont\subset\deepidenscont\not\supset\ocont
\quad\mbox{and}\quad\idenscont\not\subset\ocont.
\end{equation}
Moreover, the containment in (\ref{eq:inocontainments}) is proper 
\cite{CL:Examples}. To give some idea of just how delicate the situation is,
note the following lemma \cite{CL:Analytic}.

\lemma{lem:Cinfinityconvex}{ There exists a convex \cinfinity\
function that is not deep-\idens\ continuous.}

To proceed further toward the proof of a theorem similar to 
Theorem \ref{thm:dfirstcat}, some more definitions must be introduced.

If $A$ is a measurable subset of \reals, then its measure is denoted
\measure{A}. A set of the form $\Cup_{n\in\smallnats}[a_n,b_n]$ or 
$\Cup_{n\in\smallnats}(a_n,b_n)$ is known as a {\em right interval set},
if $b_n>a_n>b_{n+1}>0$ for all $n\in\nats$ and $a_n\to0$. The definition of a
{\em left interval set} is obvious. Any set which is the union of a right and
left interval set is just called an {\em interval set}. The following
lemmas
give useful techniques for constructing \idens\ open sets
\cite{CL:Baire} \cite[Theorem 2]{Wilczynski:CatAn}.

\lemma{lem:intlemma}
{If  $B = \bigcup_{n\in\smallnats} (a_n,b_n)$ is a right interval
set\index{interval set}\index{interval set>right}\index{right interval set}
and there exists a positive number $c$ such that
\[
\frac{b_n - a_n}{b_n}>c,
\]
for every $n\in\nats$, then $0$ is not an \I-dispersion
point\index{\I-dispersion point} of $B$.}

\lemma{lem:bninterval}{
If \, $\bigcup_{n\in\smallnats} [a_n,b_n]$ is a right interval
set with
\[
\lim_{n\to\infty}\frac{(b_n - a_n)}{b_n} = 0,
\]
then there exists an increasing sequence $\{n_m\}_{m\in\smallnats}$ of natural
numbers such that $0$ is 
an \idisp\ point of
\[
\bigcup_{m\in\smallnats}[a_{n_m},b_{n_m}].
\]
}



\theorem{thm:idensincont}{Let ${\cal C}_{\cal I}$ denote the class of all continuous
functions $f\colon[0,1]\to\reals$ which have at least one point of
\idens\  continuity. 
Then ${\cal C}_{\cal I}$ is a first category 
subset of ${\cal C}\left([0,1]\right)$.
}

Proof. We will show that there exists a dense \Gdelta\ subset $E$
of ${\cal C}$ = ${\cal C}\left([0,1]\right)$ such that every $f\in E$ is
nowhere \idens\  continuous.

For every $n\in\nats$ denote by $D_n$ the set of all $f\in{\cal C}$ such that
for every $i = 1,2,\ldots, 2^n$, $f$ is linear and nonconstant on every
interval $[(i - 1)2^{-n}, i2^{-n}]$. Note that $D_{n+1}\supset D_n$ for every
$n\in\nats$ and
\[
D = \bigcup_{n\in\smallnats}D_n
\]
is a dense subset of ${\cal C}$.

For $f\in{\cal C}$ define
\[
\|f\|_n = \max_{i = 1,2,\ldots,2^n}\left\vert f(i2^{-n}) - f((i -
1)2^{-n})\right\vert.
\]
We claim that for each open set $U$ in ${\cal C}$, there exists an $n\in\nats$
and a function $f\in D_n$ such that the ball in ${\cal C}$ centered at $f$ of
radius $\|f\|_n$ is entirely contained in $U$. To see this, first find an
$m\in\nats$ and an $f\in D_m$ such that $f\in U$. Since $U$ is open, there is a
$\delta > 0$ such that the open ball of radius $\delta$ centered at $f$ is
contained in $U$. Using the uniform continuity of $f$, we can find an $n > m$
such that if $\vert x - y\vert < 2^{-n}$, then $\vert f(x) - f(y)\vert <
\delta$. From this it is clear that $f\in D_n$ and $\|f\|_n < \delta$. The
claim becomes evident.

We now start the construction of the \Gdelta\ set $E$ as an intersection
of dense open sets $W_k$.

Let $k \ge 1$ be an integer and let $U$ be a nonempty open subset of ${\cal
C}$. Choose $f$ and 
$n\geq k$ as above. For $j = 0,1,2,\ldots,2^{n + 1}$, define
\[
g\left(\frac{j}{2^{n + 1}}\right) = 
f\left(\frac{j}{2^{n + 1}}\right).
\]

If \[
i2^{-n}\le j2^{-n-1} < (j + 1)2^{-n-1}\le(i + 1)2^{-n},
\]
where $i\in\{0,1,2,\ldots,2^n - 1\}$, put
\[
L_i = (i2^{-n}, (i + 1)2^{-n}),
\]
\[
M_j = (j2^{-n-1}, (j + 1)2^{-n-1})
\]
and let $K_j = [a_j,b_j]$ be an interval centered in $M_j$ such that
\[
\frac{\measure{K_j}}{\measure{M_j}} = 1 - \frac{1}{2^n} = 
\frac{2\measure{K_j}}{\measure{L_i}}.
\]

Let us choose $I^0_j = [c_j,d_j]$ centered in the interval $f(M_j)$ and such
that
\[
\frac{\measure{I^0_j}}{\measure{f(M_j)}} = \frac{1}{2^n}.
\]
Define the function $g$ to be linear on each of the intervals
\[
[j2^{-n-1},a_j], [a_j,b_j], \mbox{ and } [b_j, (j + 1)2^{-n-1}]
\]
in such a way that
\[
g\left([a_j,b_j]\right) = [c_j,d_j] = I^0_j.
\]
(See Figure 1.)

\putepsf{4.8}{3.86}{oldpic.ps}{The function $g(x)$}{fig:theorem23}


%\begin{figure}[hbt]
%\vbox to 4.329in{
%     \hrule width 5.127in height 0pt depth 0pt
%     \vfill
%    \special{pictfile oldpic.ps%figure.eps
%}}
%    \caption{The function $g(x)$.}
%\end{figure}






Thus, if
\[
J_j = f(M_j) = g(M_j),
\]
then
\[
\frac{\measure{g(K_j)}}{\measure{g(M_j)}} = 
\frac{\measure{I^0_j}}{\measure{J_j}} = \frac{1}{2^n}
\]
and
\[
\frac{\measure{\inv{g}(I_j^0)}}{\measure{\inv{g}(J_j)}} = 
\frac{\measure{K_j}}{\measure{M_j}} = 1 - \frac{1}{2^n}.
\]
Note that $g$ is contained in the open ball centered at $f$ of radius
$\|f\|_n$. Thus, $g\in U$.

Let $W^k_U$ be the open ball centered at $g$ of radius
\begin{eqnarray}\label{eqn:epsilon}
\e_k = 2^{-n-1}\min_{i=1,2,\ldots,2^n}\left\vert f\left(\frac{i}{2^n}\right) -
f\left(\frac{i - 1}{2^n}\right)\right\vert > 0.
\end{eqnarray}
Obviously
\[
W_k = \Cup\left\{ W_U^k\colon U \mbox{ is open and nonempty in } {\cal
C}\right\}
\]
is open and dense in ${\cal C}$, so that
\[
E = \Cap_{k\in\smallnats} W_k
\]
is a residual set in ${\cal C}$. We will show that if $h\in E$ then $h$ is
nowhere \idens\  continuous.

Let $x\in[0,1]$ be arbitrary. We will choose intervals $I_m$, 
$m\in\nats$, such
that $h(x)$ is an \idisp\/ point 
of $\bigcup_{m\in\smallnats} I_m$, but $x$ is
not an \idisp\/ point of $\inv{h}\left(\bigcup_{m\in\smallnats} I_m\right)$.
This will prove that $h$ is not \idens\  continuous at $x$.

%\vbox to 5.4in{
%     \hrule width 6.00in height 0pt depth 0pt
%     \vfill
%    \special{pictfile figure.eps}}

Let $m\in\nats$. We have $h\in W_m$, so there exists a set $U$, open in ${\cal
C}$, such that $h\in W_U^m$. Let $g$ be the center of $W_U^m$. Let $n\ge m$ be
the number given in the construction of $W^m_U$. Let $i\in\{0,1,2,\ldots,2^{n}
- 1\}$ be such that $x\in[i2^{-n},(i + 1)2^{-n}]$.


Put
\[
L_m = [i2^{-n}, (i + 1)2^{-n}].
\]
Let
\[
M^1 = \left( (2i) 2^{-n-1}, (2i + 1) 2^{-n-1}\right),
\]
\[
M^2 = \left( (2i + 1)2^{-n-1}, 2(i + 1)2^{-n-1}\right),
\]
and let $M_m\in\{M^1,M^2\}$ be such that $h(x)\notin g(M_m)$. Put $J_m =
g(M_m)$ and let $I_m^0 = [c_j,d_j]$, $K_m = [a_j,b_j]$ be as in the
construction of $g$.

Thus we have
\[
\frac{\measure{I_m^0}}{\measure{J_m}} = 
\frac{1}{2^n}\le\frac{1}{2^m} \mbox{ and }
\frac{\measure{K_m}}{\measure{M_m}}
= 1 - \frac{1}{2^n} \ge 1 - \frac{1}{2^m}.
\]
Define
\[
I_m = [c_j - \e_m, d_j + \e_m].
\]
As $h(x)\notin J_m$, we can choose a subsequence
$\{I_{m_i}\}_{i\in\smallnats}$ of $\{I_m\}_{m\in\smallnats}$ such that the union
of all intervals in the sequence $\{I_{m_i}\}_{i\in\smallnats}$ 
is a left or right interval set at $h(x)$. Without loss of
generality we may assume that it is a right interval
set at
$h(x)$. As, for each $i\in\nats$, $I_{m_i}$ and $J_{m_i}$ have a common
center and
\[
\lim_{i\to\infty}\frac{\measure{I_{m_i}}}{\measure{J_{m_i}}} = 0,
\]
Lemma \ref{lem:bninterval} says that
we can choose a subsequence $\{I_{m_{i_j}}\}_{j\in\smallnats}$ of
$\{I_{m_i}\}_{i\in\smallnats}$ such that $h(x)$ is an \idisp\/
point of
$\bigcup_{j\in\smallnats} I_{m_{i_j}}$.

On the other hand, by the way $\e_n$ was chosen in (\ref{eqn:epsilon}),
$K_n\subset \inv{h}(I_n)$. Thus, using Lemma \ref{lem:intlemma}, the fact that $x\in
L_m$ for every $m\in\nats$ and
\[
\lim_{j\to\infty}\frac{\measure{K_{n_{i_j}}}}{ \measure{L_{n_{i_j}}}} =
\lim_{j\to\infty}\frac{\measure{K_{n_{i_j}}}}{2\measure{M_{n_{i_j}}}} =
\frac{1}{2} > 0
\]
we conclude that $x$ is not an \idisp\/ point 
of $\Cup_{j\in\smallnats}
K_{n_{i_j}}$. Thus $x$ is not an \idisp\/ point of
$\inv{h}\left(\bigcup_{j\in\smallnats}I_{n_{i_j}}\right)$. This finishes the proof
of Theorem \ref{thm:idensincont}.

\section{Comparisons of \idenscont\ and \deepidenscont\ to Themselves}

Recall that a function $f\RtoR$ is in the class \bstar, if for each nonempty
perfect set $P$ there exists an open interval $I$ such that $I\cap
P\ne\emptyset$ and the restricted function $f|_{I\cap P}$ is continuous
\cite{RO:Baire*1}. It is clear from the definition that any $f\in\bstar$ must be
continuous at each point of a dense open set. This useful property is true of
the functions in \deepidenscont\ \cite{CL:Baire}.

\theorem{thm:CnnB*}{$\deepidenscont\subset\bstar$}

\theorem{thm:hereditary}{The spaces \deepidenscont\ and 
\idenscont,
equipped with the
topology of uniform convergence, 
are of the first category in themselves.}

Proof. We prove this only for the class 
\deepidenscont\ as the other case
is essentially the same.

Let $\{I_n\}_{n\in\smallnats}$ be the sequence of all 
open intervals with rational endpoints and let $C_n$ be the family of all
deep \idens\ continuous functions 
that are continuous on $I_n$ in the 
ordinary sense. By Theorem \ref{thm:CnnB*},
$\deepidenscont = \UoverN C_n$. 
Also, it is evident that the sets $C_n$ are
closed in the uniform convergence topology. Finally, for any function
$f\in C_n$ and any of its neighborhoods 
$U\subset\deepidenscont$, it is easy to 
slightly modify a function $g$ such as in 
Lemma \ref{lem:Cinfinityconvex} in such a way that
$g\in U\setminus C_n$. 
Thus, the sets $C_n$ are nowhere dense.

\bibliographystyle{plain}
%\bibliography{category}

\begin{thebibliography}{10}

\bibitem{Bruckner:DiffReal}
A.~M. Bruckner.
\newblock {\em Differentiation of Real Functions}.
\newblock Lecture Notes in Mathematics 659. Springer-Verlag, 1978.

\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:SpDensCont}
Krzysztof Ciesielski and Lee Larson.
\newblock The space of density continuous functions.
\newblock {\em Acta Math. Hung.}, to appear.

\bibitem{CL:Examples}
Krzysztof Ciesielski and Lee Larson.
\newblock Various continuities with the density, {$\cal {I}$}-density and
  ordinary topologies on {\reals}.
\newblock {\em Real Anal. Exchange}, 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{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{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{Wilczynski:CatAn}
W.~Wilczy{\'n}ski.
\newblock A category analogue of the density topology, approximate continuity,
  and the approximate derivative.
\newblock {\em Real Anal. Exchange}, 10:241--265, 1984-85.

\end{thebibliography}


\end{document}