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\firstpagenumber{102}
\received{August 25, 1993}

\Volume{20}
\IssueNumber{1}
\Year{1994/95}


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\title{Density continuous transformations on $\real^2$}

\author{Krzysztof Ciesielski,\thanks{This author was partially
supported by West Virginia University International Programs Office.}\ \
Department of Mathematics,
West Virginia University,
Morgantown, WV 26506-6310, USA
(kcies@wvnvms.wvnet.edu)
\and
W{\l}adys{\l}aw Wilczy\'{n}ski,\
Institute of Mathematics,
University of {\L}\'{o}d\'{z},
ul. Banacha 22, 90-238 {\L}\'{o}d\'{z}, Poland,
(wwil@plunlo51.bitnet)}

\date{}

\markboth{K.~Ciesielski and W.~Wilczy\'{n}ski}{Density
Continuous Transformations on $\real^2$}

\keywords{density continuity, strong density continuity}
\MathReviews{Primary: 26B05, 26A15}


\begin{document}\maketitle

\begin{abstract}

In the paper we study transformations from $\real^2$ into $\real$
and from $\real^2$ into $\real^2$ continuous with respect
to different density topologies on the domain and range.
In the former case the range will always be equipped with the
one-dimensional density topology and the domain with either
ordinary density or strong density topology. In the later case
the domain and the range will be equipped simultaneously
with the ordinary density or strong
density topology. We will investigate the relations between
these classes and with the class of ordinary continuous
transformations.
We will also examine relation between the (strong)
density continuity of $f\colon\real^2\to\real$ and of
$f(\cdot,y)$, $f(x,\cdot)$. Similar question will be considered
for transformations
$F=(f,g)\colon\real^2\to\real^2$
and their coordinate functions $f$ and $g$.
\end{abstract}

\section{Preliminaries}

The notation used throughout this paper is standard.
In particular, $\real$
or $\real^1$ stands for
the set of real numbers and $\real^2$ for the plane.
For $A\subset\real^2$ its outer two-dimensional
Lebesgue measure is denoted by
$\mtwo{A}$. Similarly, $\mone{A}$ stands for the
outer one-dimensional
Lebesgue measure of $A\subset\real$.

Recall that $x\in\real$ is a density point of
$A\subset\real$ if its density
\[
d_1(A,x)=
\lim_{\e\to 0^+}\frac{\mone{A\cap(x-\e,x+\e)}}{\mone{(x-\e,x+\e)}}=1;
\]
$(x,y)\in\real^2$ is an (ordinary) density point of
$A\subset\real^2$ if
\[
d_2(A,(x,y))=\lim_{\e\to 0^+}
\frac{\mtwo{A\cap[(x-\e,x+\e)\times(y-\e,y+\e)]}}
{\mtwo{(x-\e,x+\e)\times(y-\e,y+\e)}}=1;
\]
and $(x,y)\in\real^2$ is a strong density point of
$A\subset\real^2$ if
\[
d_s(A,(x,y))=\lim_{a\to 0^+,\, b\to 0^+}
\frac{\mtwo{A\cap[(x-a,x+a)\times(y-b,y+b)]}}{\mtwo{(x-a,x+a)
\times(y-b,y+b)}}=1.
\]
(Compare Saks
\cite[pages 106, 128]{Saks}.) Recall also that a point
$p$ is a (strong)
dispersion
point of a set $B$ if it is a (strong) density point
of the complement of $B$.
The strong density topology ${\cal T}_{\cal S}^2$ on
$\real^2$ is defined as
the family of all measurable subsets $A$ of $\real^2$
such that every
$a\in A$ is a
strong density point of $A$ \cite{GNN}.
Similarly we define density topologies
${\cal T}_{\cal N}^1$ and ${\cal T}_{\cal N}^2$
on $\real$ and $\real^2$, respectively,  using the notions of
density point on
$\real$ and ordinary density point on $\real^2$.
(Compare  \cite{GNN} and \cite{LMZ:FineTop}.)
We will drop the
superscript in this notation and write ${\cal T}_{\cal N}$ in all cases
when the space is fixed. Notice also that
${\cal T}_{\cal S}^2\subset{\cal T}_{\cal N}^2$ and recall that
the sets open in these topologies are measurable.
The symbol ${\cal T}_{\cal O}$ stands for the ordinary topology on
$\real$ or on $\real^2$.

The class of all functions from $\real^2$ to $\real$ with the density
topology ${\cal T}_{\cal N}^1$
on the range and either ordinary density topology ${\cal T}_{\cal N}^2$
or strong density topology ${\cal T}_{\cal S}^2$
on the domain will be denoted by $\cnone$ and $\csone$, respectively.
They will be termed
density and strongly density continuous transformations, respectively.
Notice that $\csone\subset\cnone$.

The class of all functions from $\real^2$ to $\real^2$ which are
continuous with
respect to
${\cal T}_{\cal N}$ (${\cal T}_{\cal S}^2$, respectively)
on the domain and the range
will be denoted by $\cntwo$ ($\cstwo$, respectively).
Functions belonging to
$\cntwo$ and $\cstwo$ are called density continuous
and strongly density
continuous transformations, respectively.

The class of all ordinarily continuous
functions from $\real^n$ to $\real^m$ will be denoted by $\co$.

Let us notice that the topologies ${\cal T}_{\cal N}^2$ and
${\cal T}_{\cal S}^2$
are invariant
under translations and exchange of coordinates. Thus,
$C(x,y)=(y,x)$ and the
translations $T_{(s,t)}(x,y)=(x+s,y+t)$
are both density and strongly density continuous.

Functions from $\real$ to $\real$ continuous with respect to the density
topology on
the domain and the range are called density continuous.
The next proposition lists some useful properties of these functions.

\prop{prop:basic}{ If $f\colon\real\to\real$ is density continuous, then
\begin{description}
\item[$\mathrm{(a)}$] $f$ is Baire 1 function; in particular,
$f$ is measurable;

\item[$\mathrm{(b)}$] $f$ has the Darboux property;

\item[$\mathrm{(c)}$] if $f$ is not constant, then there
exists $p\in f[\real]$ such that $\mone{f^{-1}(p)}=0$.
\end{description}
}

\proof (a) and (b) can be found in \cite{Bruckner:DiffReal},
if we recall that
density continuous functions from $\real$ to $\real$ are
approximately continuous
\cite{GW:AppContTrans}.

(c) follows from (a) and (b) since $\{f^{-1}(p)\colon p\in f[\real]\}$
is a partition
of $\real$ into uncountable many measurable sets.\qed

In what follows $|v|$ stands for the length of
a vector $v\in\real^n$.
For an ordinary open subset $U$ of $\real^n$ we say that
$F\colon U\to\real^n$ is
{\em bi-Lipschitz with constant $L$} ($L\geq 1$), if for every
$q,r\in U$
\[
L^{-1}|q-r|\leq|F(q)-F(r)|\leq L|q-r|.
\]
Transformation $F\colon\real^n\to\real^n$ is {\em locally
bi-Lipschitz} if for every point $p\in\real^n$ there is
$U\in{\cal T}_{\cal O}$ containing $p$
such that the restricted transformation
$F|_U$ is bi-Lipschitz. We say that
$f\colon\real^2\to\real$ is {\em locally
bi-Lipschitz} if for every point $p\in\real^2$ there is
an open rectangle $U=(a,b)\times(c,d)$ containing $p$ and
a constant $L\geq 1$ such that for every $x_0\in(a,b)$
and $y_0\in(c,d)$ the coordinate functions:
$g_{y_0}\colon(a,b)\to\real$, $g_{y_0}(x)=f(x,y_0)$, and
$h_{x_0}\colon(c,d)\to\real$, $h_{x_0}(y)=f(x_0,y)$,
are bi-Lipschitz with constant $L$.

Recall also the following facts.

\prop{prop:prelA}{ Locally convex functions from $\real$ to $\real$
are density continuous. In particular, analytic and piecewise linear
functions from $\real$ to $\real$ are density continuous. }

\proof See \cite{CL:Space}. \qed

\prop{prop:prelLip}{ Locally bi-Lipschitz transformations
from $\real^n$ into $\real^n$ are density continuous.}

\proof See \cite{Buczo:Lip}. \qed

We will finish with the following easy proposition.

\prop{prop:lattice}{ If $f,g\colon\real^n\to\real$, $n=1,2,\ldots$, are
density (strongly density)
continuous, then $\max\{f,g\}$ and $\min\{f,g\}$ are
density (strongly density) continuous.}

\proof The proof of this proposition is precisely the same as
the one for the density continuous
functions from $\real$ to $\real$. (See \cite[Theorem 2]{CL:LevelSets} or
\cite[Theorem 1(iii)]{KC:Topol}.) \qed



\section{Transformations from $\real^2$ into $\real$.}

In this section we investigate the relation between (strong)
density continuity of transformations $f\colon\real^2\to\real$
and density continuity of their sections
$f(x_0,\cdot)$ and $f(\cdot,y_0)$.
We will also study the relations between $\cnone$, $\csone$ and $\co$.
We start with the following easy fact.

\prop{prop1}{ Let $h\colon\real\to\real$
and define $f\colon\real^2\to\real$ by $f(x,y)=h(y)$.
The following conditions are equivalent.
\begin{description}
\item[$\mathrm{(a)}$] $h$ is density continuous.

\item[$\mathrm{(b)}$] $f$ is density continuous.

\item[$\mathrm{(c)}$] $f$ is strongly density continuous.
\end{description}
}

\proof Let $A\in{\cal T}_{\cal N}^1$. Then
$f^{-1}(A)=\real\times h^{-1}(A)$.
It is easy to see that
$h^{-1}(A)\in{\cal T}_{\cal N}^1$ if and only if
$\real\times h^{-1}(A)\in{\cal T}_{\cal N}^2$ if and only if
$\real\times h^{-1}(A)\in{\cal T}_{\cal S}^2$. \qed

The next theorem will be fundamental
in constructing most of the examples.

\thm{th:Lip}{ If $f\colon\real^2\to\real$ is
locally bi-Lipschitz, then $f$ is strongly density continuous.}

\proof Let $(x_0,y_0)\in\real^2$. We will show that $f$ is
strongly density continuous at $(x_0,y_0)$.
Without loss of generality we may assume that $(x_0,y_0)=(0,0)$
and that $f(0,0)=0$. Moreover, assume that
$L\geq 1$ is such that for every $x_0,y_0\in(-1,1)$
the coordinate functions:
$g_{y_0}\colon(-1,1)\to\real$, $g_{y_0}(x)=f(x,y_0)$, and
$h_{x_0}\colon(-1,1)\to\real$, $h_{x_0}(y)=f(x_0,y)$
are bi-Lipschitz with constant $L$.

Let $A\subset\real$ be measurable such that $0\not\in A$ and $0$
is a dispersion point of $A$. It is enough to prove that
$(0,0)$ is a strong dispersion point of $f^{-1}(A)$.
Let $\e>0$. Since, by Proposition \ref{prop:prelLip}, functions
$g_0$ and $h_0$ are density continuous, $0$ is a dispersion point of
$g_0^{-1}(A)$ and $h_0^{-1}(A)$. In particular,
there exists $\d_0>0$
such that for every $\d\in(0,\d_0)$
\begin{equation}\label{conLip2}
\mone{g_0^{-1}(A)\cap(-\d,\d)}<2\d\,\frac{\e}{8L^4}
\ \ \&\ \
\mone{h_0^{-1}(A)\cap(-\d,\d)}<2\d\,\frac{\e}{8L^4}.
\end{equation}
We will show that for every rectangle $R=(-a,a)\times(-b,b)$ with
$0<a,b<\d_0/(2L^2)$ we have
\[
\mtwo{f^{-1}(A)\cap R}\leq\e\,\mtwo{R}.
\]
This will finish the proof.

So choose a rectangle $R=(-a,a)\times(-b,b)$ with $0<a,b<\d_0/(2L^2)$.
We assume that $a\leq b$, the other case being similar.
In the calculation that follows we will use the Fubini
Theorem and the fact
that for every bi-Lipschitz function $\psi\colon\real\to\real$
with constant $L$ and for every measurable set $B\subset\real$
we have $\mone{\psi^{-1}(B)}\leq L\,\mone{B}$.
\begin{eqnarray*}
\mtwo{f^{-1}(A)\cap R}
&   =  &  \int_{-a}^a \mone{h_x^{-1}(A)\cap(-b,b)}\,dx\\
& \leq &  \int_{-a}^a \mone{h_x^{-1}(A\cap(h_x(0)-Lb,h_x(0)+Lb))}\,dx\\
& \leq &  \int_{-a}^a L\,\mone{A\cap(g_0(x)-Lb,g_0(x)+Lb)}\,dx\\
&   =  &
\int_{-a}^a L\,\mone{g_0\circ g_0^{-1}(A\cap(g_0(x)-Lb,g_0(x)+Lb))}\,dx\\
& \leq &  \int_{-a}^a L^2\,\mone{ g_0^{-1}(A)\cap(x-L^2b,x+L^2b)}\,dx\\
& \leq &  2a\,L^2\,\mone{ g_0^{-1}(A)\cap(-a-L^2b,a+L^2b)}
                \ \ \ \text{(as $a\leq b$)}\\
& \leq &  2a\,L^2\,\mone{ g_0^{-1}(A)\cap(-2L^2b,2L^2b)}
                \ \ \ \ \ \ \ \ \ \ \ \ \ \text{(by (\ref{conLip2}))}\\
& \leq &  2a\,L^2\,4L^2b\,\frac{\e}{8L^4}\\
&   =  &  \e\,\mtwo{R}.
\end{eqnarray*}
This finishes the proof of Theorem \ref{th:Lip}. \qed

Now we are ready to examine the relations between
the density continuity of transformations $f\colon\real^2\to\real$
and density continuity of their sections
$f(x_0,\cdot)$ and $f(\cdot,y_0)$.
We will start with the following example.

\examp{ex:coordA}{ There exists ordinary continuous function
$f\colon\real^2\to\real$ with density continuous sections
$f(x_0,\cdot)$ and $f(\cdot,y_0)$ for all $x_0,y_0\in\real$ which
is neither density nor strongly density continuous. }

\proof Choose decreasing sequence $\{a_n\}_{n=0}^\infty$ of
positive numbers converging to $0$ such that
$0$ a dispersion point of $A=\bigcup_{n=0}^\infty[a_{2n+1},a_{2n}]$.
(For example put $a_{2n+1}=[(n+5)!]^{-1}$
and $a_{2n}=[(n+5)!]^{-1}+[(n+6)!]^{-1}$.)
Define $g\colon\real\to\real$ by putting $g(0)=0$, $g(a_n)=a_{n+1}$
for all $n\in\natural$
and extend it to an increasing function
linearly on each of the intervals $(-\infty,0]$,
$[a_0,\infty)$ and $[a_{n+1},a_n]$ for all $n\in\natural$.
It is easy to see that $g$ is ordinary continuous. However,
it is not density continuous, since $0$ is a density point of
$\real\setminus A$, while
$0=g^{-1}(0)$ is not a density point of $g^{-1}(\real\setminus A)$,
since
$g^{-1}(\real\setminus A)\cap(0,a_0]\subset A$.

Now, for $y\leq x$ define
\[
f(x,y)= \left\{
\begin{array}{cl}
  0
     & y\leq x/3\\
\frac{6g(x)}{x}\cdot y - 2g(x)
     & 0<x/3\leq y\leq x/2\\
g(x)
     & x/2\leq y \\
\end{array}\right.
\]
and define $f(x,y)=f(y,x)$ for $x\leq y$.
Notice that for fixed $x>0$ the function $f(x,\cdot)$ is linear
for $x/3\leq y\leq x/2$.

It is easy to see that $f\colon\real^2\to\real$ is continuous.
Also, sections
$f(x_0,\cdot)$ and $f(\cdot,y_0)$ of $f$ are density continuous
for all $x_0,y_0\in\real$, since they are piecewise
density continuous.
(They are either constant or bi-Lipschitz; see
Propositions \ref{prop:prelA}
and \ref{prop:lattice}.)

To finish the proof it is enough to show that $f$ is not density
continuous. To see this, notice that
\[
f^{-1}(A)\supset\{(x,y)\colon 0< x/3\leq y\leq x/2\}
\cap\left[ \bigcup_{n=0}^\infty[a_{2n+2},a_{2n+1}]\times\real\right]=V
\]
and that (ordinary) density of $V$
at $(0,0)$ is $1/48$, since
the
ordinary density of $\{(x,y)\colon 0< x/3\leq y\leq x/2\}$
at $(0,0)$ is
$1/48$ and
$0$ is a right density point of
$\bigcup_{n=0}^\infty[a_{2n+2},a_{2n+1}]$.
Thus, $(0,0)$ is not a dispersion point of $f^{-1}(A)$
while $0=f(0,0)$ is a dispersion point of $A$. \qed

Example \ref{ex:coordA} shows that we cannot conclude density
continuity of function $f\colon\real^2\to\real$
from the density continuity of its sections.
What about the other way around? Can we conclude density continuity
of sections of $f\colon\real^2\to\real$ if $f$ is either
density continuous or strongly density continuous?
The next example shows that the answer for
this question is NO, in case of density continuity of $f$.
Then we will show that the answer is
YES in case when $f$ is strongly density continuous.

\examp{ex:coordB}{ There exists continuous and density
continuous transformation
$f\colon\real^2\to\real$ such that
$f(\cdot,0)$ is not density continuous. }

\proof Define $f$ by
\[
f(x,y)= \left\{
\begin{array}{cl}
0        & x\leq 0\\
0        & x>0\ \text{ and } |y|>x^2\\
(1-\frac{|y|}{x^2})g(x)
         & x>0\ \text{ and } |y|\leq x^2,\\
\end{array}\right.
\]
where $g$ is as in Example \ref{ex:coordA}. It is easy to see that
$f$ is continuous.
$f$ is density continuous at points $\neq(0,0)$ by Theorem \ref{th:Lip}
and Proposition \ref{prop:lattice}.
It is density continuous at $(0,0)$, because $f$ equals to zero
on a set
$D=\{(x,y)\colon x\leq 0\text{ or }x>0\ \text{ and } |y|>x^2\}$
and $(0,0)$ is (ordinary) density point of $D$.
Finally, $f(\cdot,0)$ is not density continuous, since
it is equal to $g$ on
$[0,\infty)$ and $g$ is not right
density continuous at $0$. \qed

Now we will show that density continuity of
$f(x_0,\cdot)$ and $f(\cdot,y_0)$ follows from the
strong density continuity of $f\colon\real^2\to\real$. The argument
is essentially the same as in the proof that
the sections of strongly approximately continuous functions
from $\real^2$ into $\real$ are approximately continuous
\cite[p. 502]{GNN}.
For the convenience of the reader we will extract here the main parts
of that proof in the form of two lemmas. They will imply both of
these results.

\lem{lem:A}{ Let $(X,\sigma)$ be a topological space, $y_0\in X$
and let $\tau$ be a topology on $X\times X$ such that the following
condition holds:
\begin{description}
\item[$\mathrm{(CL)}$] for every $U\in\tau$ and $x_0\in X$
\[
\text{if }\ x_0\in\cl_\sigma\{x\colon(x,y_0)\in U\},
\text{ then }\ (x_0,y_0)\in\cl_\tau U.
\]
\end{description}
Then for every regular topological space $Y$ and
continuous $f\colon(X\times X,\tau)\to Y$
function $g\colon(X,\sigma)\to Y$ defined by
$g(x)=f(x,y_0)$ is continuous. }

\proof Let $x_0\in X$. We will show that $g$ is continuous at $x_0$.
So let $V\subset Y$ be open neighborhood of $z_0=g(x_0)=f(x_0,y_0)$.
It is enough to prove that $x_0$ is a $\sigma$-interior
point of $g^{-1}(V)$.

By way of contradiction assume that it is not the case. Then
$x_0\in \cl_\sigma g^{-1}(Y\setminus V)$. By the regularity of
$Y$ we can find disjoint open sets $U,W\subset Y$ such that
$z_0\in W$ and $Y\setminus V\subset U$. In particular,
\[
x_0\in \cl_\sigma g^{-1}(U)=
\cl_\sigma\{x\colon (x,y_0)\in f^{-1}(U)\}.
\]
Hence using (CL) to $f^{-1}(U)\in\tau$, we obtain
\[
(x_0,y_0)\in\cl_\tau f^{-1}(U)\subset
\cl_\tau f^{-1}(Y\setminus W)
\]
which contradicts the fact that $(x_0,y_0)$ is a $\tau$-interior
point of $f^{-1}(W)$. \qed

\lem{lem:B}{ $(\real,{\cal T}_{\cal N}^1)$ and
$(\real^2,{\cal T}_{\cal S}^2)$
satisfy condition $\mathrm{(CL)}$ of Lemma
$\mathrm{\ref{lem:A}}$ for every $y_0\in\real$,
i.e.,
\begin{description}
\item[$\mathrm{(CL)}$] for every $U\in{\cal T}_{\cal S}^2$ and
$x_0,y_0\in\real$
\[
\text{if }\ x_0\in\cl_{{\cal T}_{\cal N}^1}\{x\colon(x,y_0)\in U\},
\text{ then }\ (x_0,y_0)\in\cl_{{\cal T}_{\cal S}^2} U.
\]
\end{description}
}

\proof The proof of this Lemma
is implicitly contained in the proof of
\cite[Theorem 4]{GNN}. For completeness sake we will sketch it here.

So let $U\in{\cal T}_{\cal S}^2$ and $x_0,y_0\in\real$ be such that
$x_0\in\cl_{{\cal T}_{\cal N}^1}V$, where
$V=\{x\colon(x,y_0)\in U\}$. Then there exists $\e>0$
and a sequence of intervals $I_n=(a_n,b_n)$ centered at $x_0$ such that
$\lim_{n\to\infty} (b_n-a_n)=0$ and
\begin{equation}\label{conAAA}
\frac{\mone{V\cap I_n}}{\mone{I_n}}>\e\ \text{ for all }\ n\in\natural.
\end{equation}
Now, fix $n\in\natural$ and notice that
for every $p\in V$ point $(p,y_0)$ is an interior point of $U$.
Thus, for $\e_n=\mone{I_n}$ and every $p\in V$
there exists $\d_p>0$, such that
\begin{equation}\label{conBBB}
\mtwo{U\cap R}>(1-\e_n)\,\mtwo{R}\ \text{ for every }\ R\in{\cal R}_p,
\end{equation}
where ${\cal R}_p$ is a family of all rectangles
$(p-\d,p+\d)\times(y_0-\d^\prime,y_0+\d^\prime)$
with $\d,\d^\prime\in(0,\d_p)$. Notice that for
$\V_p=\{(p-\d,p+\d)\subset I_n\colon 0<\d<\d_p\}$ the collection
$\{\V_p\colon p\in I_n\cap V\}$ is a Vitali cover of $I_n\cap V$.
In particular, there exists a finite collection
$\{J_i\}_{i=1}^m$ of disjoint intervals such that
$J_i=(p_i-\d_i,p_i+\d_i)\in\V_{p_i}$ and
\begin{equation}\label{conCCC}
\mone{I_n\cap V}-m_1\left(\bigcup_{i=1}^mJ_i\right)\leq
m_1\left((I_n\cap V)\setminus\bigcup_{i=1}^m J_i\right)<\e_n^2.
\end{equation}
Now, let $\d^\prime=\min\{\d_i\colon i=1,2,\ldots,m\}$ and define
$R_n=I_n\times(y_0-\d^\prime,y_0+\d^\prime)$.
Then the rectangles $J_i^\star=J_i\times(y_0-\d^\prime,y_0+\d^\prime)$
are disjoint subsets of $R_n$, and
\begin{eqnarray*}
\mtwo{U\cap R_n}
& \geq &  \sum_{i=1}^m \mtwo{U\cap J_i^\star}
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\text{by (\ref{conBBB})})\\
& \geq &  \sum_{i=1}^m (1-\e_n)\,\mtwo{J_i^\star}\\
&   =  &  (1-\e_n)\,2\d^\prime\,
          m_1\left(\bigcup_{i=1}^m J_i\right)
        \ \ \ \ \ \ \ \ \ \ \ \ \ (\text{by (\ref{conCCC})})\\
& \geq &  (1-\e_n)\,2\d^\prime\,(\mone{I_n\cap V}-\e_n^2)
                       \ \ \ \ \ \ (\text{by (\ref{conAAA})})\\
& \geq &  (1-\e_n)\,2\d^\prime\,(\e\,\mone{I_n}-\e_n^2)\\
&   =  &  (1-\e_n)(\e-\e_n)\,2\d^\prime\,\e_n\\
&   =  &  (1-\e_n)(\e-\e_n)\,\mtwo{R_n}.
\end{eqnarray*}
Thus,
\[
\frac{\mtwo{U\cap R_n}}{\mtwo{R_n}}\geq\frac{\e}{2}
\]
for sufficiently large $n$ since $\e_n$ approaches $0$.
So $(x_0,y_0)\in\cl_{{\cal T}_{\cal S}^2} U$. \qed

As a corollary we conclude the following theorem.

\thm{th:strong}{ Let $f\colon\real^2\to\real$, $x_0,y_0\in\real$ and
$g,h\colon\real\to\real$ be defined by
$g(x)=f(x,y_0)$ and $h(y)=f(x_0,y)$ for $x,y\in\real$.
\begin{description}
\item[$\mathrm{(i)}$] If $f$ is strongly density continuous,
then $g$ and $h$ are
density continuous.
\item[$\mathrm{(ii)}$] If $f$ is strongly approximately
continuous, then $g$ and $h$ are
approximately continuous.
\end{description}
}

\proof It follows immediately from Lemmas \ref{lem:A} and \ref{lem:B} if
we notice that the ordinary topology and the density topology on $\real$
are regular. \qed

We will finish this section with the following comparison of classes
of density continuous $\cnone$, strongly density continuous $\csone$
and ordinary continuous
$\co$ functions from $\real^2$ into $\real$.

\thm{th:chart}{
\[
\begin{array}{ccccc}
   \csone     & \subset &    \cnone     &         &     \nonumber \\
    \cup      &         &     \cup      &         &     \nonumber \\
\csone\cap\co & \subset & \cnone\cap\co & \subset & \co \nonumber \\
\end{array}
\]
All the containments are proper.
}

\proof The inclusions follow immediately from
$\csone\subset\cnone$.

To see that they are proper
it is enough to show
$\csone\not\subset\co$ for vertical
inclusions and
$\co\not\subset\cnone$ and
$\cnone\cap\co\not\subset\csone$
for horizontal inclusions.

$\csone\not\subset\co$ follows from Proposition \ref{prop1}
for an arbitrary function $f(x,y)=h(y)$, where $h\colon\real\to\real$
is density continuous and not continuous. For example,
we can define
\[
h(x)= \left\{
\begin{array}{cl}
0 & x\not\in A \\
  \\
\frac{\min\{a_{2n}-x,x-a_{2n+1}\}}{a_{2n}-a_{2n+1}} &
x\in(a_{2n+1},a_{2n}), \\
\end{array}\right.
\]
where $A=\bigcup_{n=0}^\infty[a_{2n+1},a_{2n}]$
is from Example \ref{ex:coordA}.

$\co\not\subset\cnone$ follows from Proposition \ref{prop1}
for any function $f(x,y)=g(y)$, where $g\colon\real\to\real$
is continuous and not density continuous. For example, function $g$ from
Example \ref{ex:coordA} works.

$\cnone\cap\co\not\subset\csone$ is justified by function
$f$ from Example \ref{ex:coordB}
since, by Theorem \ref{th:strong}(i), $f$ cannot be strongly
density continuous.
\qed

     \begin{figure}[htb]
     \[
       \vbox to 1.74in{\hrule width 2.53in height 0pt depth 0pt \vfill
         \special{illustration FA.ps scaled 500}}
       \vbox to 1.74in{\hrule width 2.53in height 0pt depth 0pt \vfill
         \special{illustration A.ps scaled 500}}
     \]
    \caption{Sets $F^{-1}(A)$ (left) and $A$ (right) from
             Example \protect\ref{exA}\label{PictExA}}
    \end{figure}

\section{Transformations from $\real^2$ into $\real^2$.}

In this section we
consider the interrelations between (strongly) density continuous
transformations $F=(f,g)\colon\real^2\to\real^2$ and their coordinate
functions
$f,g\colon\real^2\to\real$.

If $f$ and $g$ are the coordinate functions of transformation
$F\colon\real^2\to\real^2$, then $f,g\colon\real^2\to\real$.
It might happen, however, that either $f$ or $g$ depends of only
one variable, e.g.,
$f(x,y)=h(y)$ for some $h\colon\real\to\real$. Then, according to
Proposition \ref{prop1}, we can replace $f$ with $h$ when examining
the (strong) density continuity of $f$. We will use this convention
throughout the rest
of the paper.

The next easy fact forms a base for the discussion of this section.

\thm{th:coord1}{ Let $F\colon\real^2\to\real^2$ be a transformation
with coordinate functions
$f,g\colon\real^2\to\real$, i.e., such that $F(x,y)=(f(x,y),g(x,y))$.
\begin{description}
\item[$\mathrm{(a)}$] If $F$ is strongly density continuous,
then $f$ and $g$
are also strongly
density continuous.

\item[$\mathrm{(b)}$] If $F$ is density continuous, then $f$
and $g$ are density
continuous.
\end{description}
}

\proof (a). Assume that $F$ is strongly density continuous and let
$A\in{\cal T}_{\cal N}^1$.
We have to show that $f^{-1}(A),g^{-1}(A)\in{\cal T}_{\cal S}^2$.
But $A\times\real\in{\cal T}_{\cal S}^2$ and so,
$f^{-1}(A)=F^{-1}(A\times\real)\in{\cal T}_{\cal S}^2$. Similarly,
$\real\times A\in{\cal T}_{\cal S}^2$ so that
$g^{-1}(A)=F^{-1}(\real\times A)\in{\cal T}_{\cal S}^2$.

The proof of part (b) is identical. \qed

     \begin{figure}[htb]
     \[
       \vbox to 1.74in{\hrule width 2.53in height 0pt depth 0pt \vfill
         \special{illustration FB.ps scaled 500}}
       \vbox to 1.74in{\hrule width 2.53in height 0pt depth 0pt \vfill
         \special{illustration B.ps scaled 500}}
     \]
    \caption{Sets $F^{-1}(B)$ (left) and $B$ (right) from
             Example \protect\ref{exB}\label{PictExB}}
    \end{figure}



At this moment one might be tempted to prove the
converse of (a) and (b) of Theorem \ref{th:coord1}.
Indeed, such a claim would be obvious
if either the ordinary or the strong density topology on $\real^2$
was a product
topology of the one-dimensional density topology on $\real$. This,
however, is not the case
and the next two examples show that neither implication in
(a) nor (b) of Theorem \ref{th:coord1} can be reversed.

\examp{exA}{ The transformation $F\colon\real^2\to\real^2$,
$F(x,y)=(x,y^3)$, is not
density continuous, while its coordinate functions $f(x)=x$
and $g(y)=y^3$
are density continuous. }

\proof The functions $f$ and $g$ are density continuous, since
all real analytic functions are
density continuous.
(See Proposition \ref{prop:prelA}.)
To see that $F$ is not density continuous put
$A=\{(u,v)\colon |v|>\left|u^3\right|\}\cup\{(0,0)\}$ and notice that
$F^{-1}(A)=\{(u,v)\colon |v|>|u|\}\cup\{(0,0)\}$.
(See Figure \ref{PictExA}.)
It is routine to check that $A\in{\cal T}_{\cal N}$, while
$F^{-1}(A)\not\in{\cal T}_{\cal N}$ since
$d_2(F^{-1}(A),(0,0))=1/2\neq 1$. \qed


\examp{exB}{ The transformation $F\colon\real^2\to\real^2$,
$F(x,y)=(x,x+y)$, is not
strongly density continuous, while its coordinate
functions $f(x,y)=x$ and $g(x,y)=x+y$
are strongly density continuous. }

\proof $f$ is strongly density continuous by
Proposition \ref{prop1}.
The function $g$ is strongly density continuous by
Theorem \ref{th:Lip}
since it is obviously bi-Lipschitz.


To see that $F$ is not strongly density continuous put
\begin{eqnarray*}
B & = & \{(u,v)\colon |v-u|>u^2\}\cup\{(0,0)\}\\
  & = & \{(u,v)\colon v>u+u^2\ \text{ or } v<u-u^2\}\cup\{(0,0)\}
\end{eqnarray*}
and notice that
$F^{-1}(B)=\{(u,v)\colon |v|>u^2\}\cup\{(0,0)\}$.
(See Figure \ref{PictExB}.)
It is easy to check that $B\in{\cal T}_{\cal S}^2$, while
$F^{-1}(B)\not\in{\cal T}_{\cal S}^2$ because $(0,0)$ is not
a strong density point of
$F^{-1}(B)$. \qed

Notice that the transformation $F(x,y)=(x,x+y)$ is evidently
bi-Lipschitz. In particular, the next corollary proves that
Proposition \ref{prop:prelLip} cannot be generalized to
strongly density continuous transformations from $\real^2$
into $\real^2$.

\cor{cor:StrongAndLip}{ There exists a bi-Lipschitz
transformations from $\real^2$ into $\real^2$ which is not
strongly density continuous. \qed }

Example \ref{exA} suggests that there is no real chance to
reverse implication
(b) of Theorem
\ref{th:coord1}. Also, Example \ref{exB} does no leave much hope for
reversing implication Theorem \ref{th:coord1}(b). We can still hope,
however, that
$F\colon\real^2\to\real^2$, defined by $F(x,y)=(f(x),g(y))$
is strongly density
continuous, provided $f,g\colon\real\to\real$ are density
continuous.
But, even this claim is too strong, as the precise condition
in this direction is
given by the next theorem.

\thm{th:StrongCoord}{ Let $f,g\colon\real\to\real$ and define
the transformation $H\colon\real^2\to\real^2$
by $H(x,y)=(f(x),g(y))$. If $H$ is not constant, then
$H$ is strongly density continuous if and only if
functions $f$ and $g$ are density continuous and
$\mone{f^{-1}(p)}=\mone{g^{-1}(p)}=0$
for every $p\in\real$. }

\proof ``$\Longrightarrow$'' Assume first that $H$
is strongly density continuous.
Then $f$ and $g$ are density continuous by
Theorem \ref{th:coord1} and
Proposition \ref{prop1}.

To prove the additional part first notice that neither $f$
nor $g$ is constant.
To see this assume, to the contrary, that $g$ is constant
and equal to $b\in\real$.
Then $f$ is not constant, since we assumed that $H$ is not constant.
Thus, by Proposition \ref{prop:basic}(c),
there exists $p\in f[\real]$ with $\mone{f^{-1}(p)}=0$.
Now notice that
$A=[\real\times(\real\setminus\{b\})]\cup\{(p,b)\}
\in{\cal T}_{\cal S}^2$
as a set of full measure,
while $H^{-1}(A)=H^{-1}((p,b))=f^{-1}(p)\times\real$ is
non-empty and has
two-dimensional Lebesgue measure zero. Thus,
$H^{-1}(A)\not\in{\cal T}_{\cal S}^2$.
This contradiction establishes that $g$ is not constant.
A similar argument shows that $f$ is not constant.

Now, we will show that $\mone{f^{-1}(p)}=0$
for every $p\in\real$. This will imply that
$\mone{g^{-1}(p)}=0$ for every $p\in\real$ by using the same
argument for
$G\colon\real^2\to\real^2$ defined by $G(x,y)=(g(x),f(y))$,
which is also
strongly density continuous.

So pick $p\in\real$ and, by way of contradiction, assume that
$\mone{f^{-1}(p)}>0$. Then
by the Lebesgue Density Theorem, there exists $a\in f^{-1}(p)$
with $d_1(f^{-1}(p),a) = 1$.
Since $g$ is not constant, by Proposition \ref{prop:basic}(c)
we can find $q\in g[\real]$ such that
$\mone{g^{-1}(q)}=0$. Put
$A=[(\real\setminus\{p\})\times\real]\cup\{(p,q)\}$. Then
$A\in{\cal T}_{\cal S}^2$ as a set of full measure.
On the other hand,
\[
H^{-1}(A)=[(\real\setminus f^{-1}(p))\times\real]
\cup[f^{-1}(p)\times g^{-1}(q)]
\]
does not belong to ${\cal T}_{\cal S}^2$.
To see this pick $(a,b)\in f^{-1}(p)\times g^{-1}(q)\subset H^{-1}(A)$.
Then
\begin{eqnarray*}
d_s(H^{-1}(A),(a,b))
& = & d_s((\real\setminus f^{-1}(p))\times\real,(a,b)) \\
& = & d_1(\real\setminus f^{-1}(p),a) \\
& = & 1-d_1(f^{-1}(p),a) = 1-1 \neq 1,
\end{eqnarray*}
since $\mtwo{f^{-1}(p)\times g^{-1}(q)}=0$.

The proof of implication ``$\Longrightarrow$'' is completed.

``$\Longleftarrow$'' First notice that it is enough to show that
the transformation
$F(x,y)=(f(x),y)$ is strongly density continuous under our
assumption, since $C(x,y)=(y,x)$ is strongly density continuous.
Moreover, it is enough to show that $F(x,y)=(f(x),y)$ is
strongly density continuous
at $(0,0)$, since translations are strongly density continuous.
We can also assume that $f(0)=0$.

So let $A\subset \real^2$ be such that
$(0,0)$ is not a strong dispersion point of $A$. We will finish
the proof by
showing that $F(0,0)=(0,0)$ is not a strong dispersion of $F[A]$.
In the proof we will need the following easy lemma
in which $A^x$ stands for a vertical section of
$A\subset\real^2$ given
by $x$, i.e., $A^x=\{y\colon (x,y)\in A\}$.

\lem{lem:sect}{ Let $\e\in(0,1/2)$ and let
$A\subset R=[-a,a]\times[-b,b]$ be measurable
and such that
\[
\frac{\mtwo{A}}{\mtwo{R}}\geq\e.
\]
If $B=\{x\in[-a,a]\colon \mone{A^x}\geq 2b\e^2\}$, then
$\mone{B}\geq 2a\e^2$.
}

\proof Easy, by the Fubini Theorem. \qed

Now, we can come back to the proof of Theorem \ref{th:StrongCoord}.
Since
$(0,0)$ is not a strong dispersion point of $A$, there exists
$\e\in(0,.5)$
and a sequence of rectangles $R_n=[-a_n,a_n]\times[-b_n,b_n]$
such that
$\max\{a_n,b_n\}\to 0$ and
\begin{equation}\label{conA}
\frac{\mtwo{A\cap R_n}}{\mtwo{R_n}}>3\e \text{ for all }\
n\in\natural.
\end{equation}

Now, by induction on $i\in\natural$, we define an increasing
sequence $\{n_i\}_{i=0}^\infty$ of indices,
sequences $\{c_i\}_{i=0}^\infty$, $\{d_i\}_{i=0}^\infty$
of positive numbers and a
sequence $\{B_i\}_{i=0}^\infty$ of
sets such that the following inductive conditions hold for
every $i\in\natural$
\begin{description}
\item[$\mathrm{(i)}$] $0<c_i<d_i$ and $\frac{d_i}{c_{i-1}}<
\frac{1}{i}$
for $i>0$.

\item[$\mathrm{(ii)}$]
$B_i\subset[-a_{n_i},a_{n_i}]\cap f^{-1}((-d_i,d_i)
\setminus(-c_i,c_i))$
is such that
$\mone{B_i}\geq 2a_{n_i}\e^2$ and
$\mone{A^x\cap[-b_{n_i},b_{n_i}]}\geq 2b_{n_i} \e^2$
for all $x\in B_i$.
\end{description}

To make the inductive step take $i\in\natural$
such that the construction is already done for all natural
numbers
less than $i$. If $i=0$ put $d_0=1$. Otherwise, choose
$d_i\in(0,c_{i-1}/i)$.
This guarantees satisfaction of (i) for step $i$.

Now, since $0$ is a density point of $(-d_i,d_i)$ and $f$
is density continuous,
$0$ is a density point of $f^{-1}((-d_i,d_i))$ and $(0,0)$
is a strong density point of
$f^{-1}((-d_i,d_i))\times\real$.

Choose $n_i\in\natural$, $n_i>n_{i-1}$ for $i>0$, such that
\[
\frac{\mtwo{[f^{-1}((-d_i,d_i))\times\real]\cap R_{n_i}}}
{\mtwo{R_{n_i}}}>1-\e.
\]
Then, by (\ref{conA}),
\begin{equation}\label{conB}
\frac{\mtwo{A\cap [f^{-1}((-d_i,d_i))\times\real]\cap R_{n_i}}}
{\mtwo{R_{n_i}}}> 2\e.
\end{equation}
Moreover, since
\[
m_2\left(\left[
\bigcap_{c>o}f^{-1}((-c,c))\times\real\right]\cap R_{n_i}\right) =
\mtwo{[         f^{-1}(0)  \times\real      ]\cap R_{n_i} } = 0
\]
we can find $c_i\in(0,d_i)$ such that
$\mtwo{ [f^{-1}((-c_i,c_i))\times\real]\cap R_{n_i} }<\e\,
\mtwo{ R_{n_i} }$, i.e.,
that
\[
\frac{ \mtwo{ [f^{-1}((-c_i,c_i))\times\real]\cap R_{n_i} } }
{ \mtwo{ R_{n_i} } } < \e.
\]
Hence, by (\ref{conB}),
\[
\frac{\mtwo{A\cap [f^{-1}((-d_i,d_i)\setminus(-c_i,c_i))
\times\real]\cap R_{n_i}}}
{\mtwo{R_{n_i}}}>\e.
\]
Now, using Lemma \ref{lem:sect} with
$A\cap [f^{-1}((-d_i,d_i)\setminus(-c_i,c_i))\times\real]
\cap R_{n_i}$ and $R_{n_i}$
we can find $B_i$ satisfying (ii).

This completes the inductive construction.

From condition (ii) it follows immediately that $0$ is not
a dispersion point
of $B=\bigcup_{i=0}^\infty B_i$. Hence, since $0\not\in B$
and $f$ is density continuous,
$0=f(0)$ is not a dispersion point of $f[B]$. So, there
 exists $\d>0$
and a decreasing sequence $\{p_n\}_{n=0}^\infty$ converging to
$0$ such that
\[
\frac{ \mone{ f[B]\cap [-p_n,p_n] } }{2 p_n}\geq 2\d\
\text{ for every }\ n\in\natural.
\]

For fixed $i\in\natural$ let $k_i$ be the smallest
natural number such that
$c_{k_i}\leq p_i$.
Then
$f[B]\cap[-p_i,p_i]=\bigcup_{j=k_i}^\infty f[B_j]\cap[-p_i,p_i]$
since $f[B_k]\subset(-d_k,d_k)\setminus(-c_k,c_k)$
for every $k\in\natural$.
Hence,
\begin{eqnarray*}
&  &
\frac{ \mone{f[B_{k_i}]\cap[-p_i,p_i]} }{2p_i}
+  \frac{2 d_{k_i + 1} }{ 2 c_{k_i} }\\
& \geq &
\frac{ \mone{f[B_{k_i}]\cap[-p_i,p_i]} }{2p_i} +
        \frac{ \mone{(-d_{k_i + 1},d_{k_i + 1})} }{ 2 p_i }\\
& \geq &
\frac{ \mone{f[B_{k_i}]\cap[-p_i,p_i]} }{2p_i} +
\frac{ m_1\left(\bigcup_{j=k_i+1}^\infty f[B_j]\cap[-p_i,p_i]\right) }
{ 2 p_i }\\
&   =  & \frac{ \mone{f[B]\cap[-p_i,p_i]} }{2p_i}\\
& \geq & 2\d
\end{eqnarray*}
for every $i\in\natural$. However, the sequence
$\{\frac{d_{i+1}}{c_i}\}$ converges to $0$.
Thus, there exists $i_0\in\natural$ such that
$\frac{d_{k_i+1}}{c_{k_i}}<\d$ for
$i\geq i_0$. In particular,
\begin{equation}\label{conC}
\frac{ \mone{f[B_{k_i}]\cap[-p_i,p_i]} }
{2p_i}>\d\ \text{ for }\ i\geq i_0.
\end{equation}

Now, notice that diameters of the rectangles
$S_i=[-p_i,p_i]\times[-b_{n_{k_i}},b_{n_{k_i}}]$
converge to $0$. We will use these rectangles
to show that $(0,0)$ is not a
strong dispersion point of $F[A]$. So
consider $F[A]\cap S_i$.
For every $v\in f[B_{k_i}]\cap[-p_i,p_i]$ and each $x\in B_{k_i}$
such that $f(x)=v$ we have
\[
(F[A]\cap S_i)^v=(F[A])^v\cap[-b_{n_{k_i}},b_{n_{k_i}}]
\supset A^x\cap[-b_{n_{k_i}},b_{n_{k_i}}].
\]
Hence, by (ii), $\mone{(F[A]\cap S_i)^v}\geq 2b_{n_{k_i}}\e^2$
for every $v\in f[B_{k_i}]\cap[-p_i,p_i]$.
Therefore, by (\ref{conC}) and Fubini Theorem,
\[
\mtwo{F[A]\cap S_i}\geq 2b_{n_{k_i}}\e^2
\mone{f[B_{k_i}]\cap[-p_i,p_i]}
\geq 2b_{n_{k_i}}\e^2 \d 2p_i = \mtwo{S_i} \d\e^2
\]
for all $i\geq i_0$, i.e.,
\[
\frac{ \mtwo{F[A]\cap S_i} }{ \mtwo{S_i} }\geq \d\e^2\
\text{ for every }\ i\geq i_0.
\]
Thus, $(0,0)$ is not a strong dispersion point of $F[A]$.

This finishes the proof of Theorem \ref{th:StrongCoord}. \qed

Notice that the above proof can easily be modified
to obtain the following local version of
Theorem \ref{th:StrongCoord}.

\thm{thm:remark}{ Let $f\colon\real\to\real$.
Then $F(x,y)=(f(x),y)$ is strongly density continuous
at $(a,b)\in\real^2$ if and only if $f$ is density continuous
at $a$ and
$a$ is a dispersion point of $f^{-1}(\{f(p)\})$. \qed}

We will finish this section with the proof that
there are no inclusion relations between
$\co$, $\cntwo$ and $\cstwo$ as stated in the theorem below.

\thm{th:incl}{
\begin{description}
\item[$\mathrm{(a)}$] $\co\cap\cntwo\not\subset\cstwo$;
\item[$\mathrm{(b)}$] $\co\cap\cstwo\not\subset\cntwo$;
\item[$\mathrm{(c)}$] $\cntwo\cap\cstwo\not\subset\co$.
\end{description}
}
\proof (a) It is justified by
$F\colon\real^2\to\real^2$ defined by $F(x,y)=(x,x+y)$
from Example \ref{exB}. It was proved there that $F$ is
not in $\cstwo$. $F$ is also evidently bi-Lipschitz, so
it is continuous and, by Proposition \ref{prop:prelLip},
it is in $\cntwo$.

(b) It is justified by
$F\colon\real^2\to\real^2$, $F(x,y)=(x,y^3)$,
since it is evidently in $\co$, $F\not\in\cntwo$
by Example \ref{exA} and $F\in\cstwo$ by Theorem \ref{th:StrongCoord},
since $f(x)=x$ and $g(y)=y^3$ are density continuous homeomorphisms.

(c) It is justified by
$F\colon\real^2\to\real^2$ defined by
$F(x,y)=(f(x),y)$, where $f(x)=x+h(x)$ and
$h$ is from Theorem \ref{th:chart}.

Since $h$ and $f$ are clearly discontinuous at $0$ we have
$F\not\in\co$.
To prove that $F\in\cstwo$ notice first that $f$ is density
continuous: at points
$x\neq 0$, since it is there piecewise linear on $\real\setminus\{0\}$,
and at $0$, since $f$ is the identity function on a set
$\real\setminus A$, for which $0$ is a density point.
Now, $F\in\cstwo$ follows from Theorem \ref{th:StrongCoord}
if we notice that none of
the slopes of the linear pieces of $h$ equals $-1$,
i.e., $f^{-1}(p)$ is at most countable for every $p\in\real$.

To see that $F\in\cntwo$ we have to consider two kinds of points.
$F$ is density continuous at points of
$\real^2\setminus (A\times\real)\in{\cal T}_{\cal N}^2$
since it is the identity mapping on this set.
$F$ is continuous at points of $A\times\real$ since
at every point of this set $F$
is either locally bi-Lipschitz (Proposition \ref{prop:prelLip})
or is a maximum (minimum) of two functions with this
property (see Proposition \ref{prop:lattice}). \qed

\begin{thebibliography}{22}
\bibitem{Bruckner:DiffReal} A. M. Bruckner,
{\it Differentiation of Real Functions,}
 Lecture Notes in Mathematics 659, Springer-Verlag 1978.

\bibitem{Buczo:Lip} Zoltan Buczolich,
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