\documentclass{rae} \usepackage{amsmath,amsthm,amssymb,epic} %\coverauthor{Krzysztof Ciesielski and Harvey Rosen} %\covertitle{Two Examples Concerning Extendable and Almost Continuous %Functions} \received{March 19, 1999} \MathReviews{Primary: 26A15; Secondary: 54A25.} \keywords{extendable functions, peripherally continuous functions.} \firstpagenumber{579} \markboth{Krzysztof Ciesielski and Harvey Rosen}{Extendable and Almost Continuous Functions} \author{Krzysztof Ciesielski% \thanks{ The work of the first author was partially supported by NSF Cooperative Research Grant INT-9600548 with its Polish part financed by Polish Academy of Science PAN. \endgraf Papers authored or co-authored by a Contributing Editor are managed by a Managing Editor or one of the other Contributing Editors.}, Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310. e-mail: {\tt K\_Cies@math.wvu.edu}\\ web page: {\tt http://www.math.wvu.edu/homepages/kcies} \and Harvey Rosen, Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487. e-mail: {\tt hrosen@gp.as.ua.edu}} \title{TWO EXAMPLES CONCERNING EXTENDABLE AND ALMOST CONTINUOUS FUNCTIONS} %%%Put Autnor's Definitons Below Here%%% \newcommand{\bbP} {\mathbb{P}} \newcommand{\bbR} {\mathbb{R}} \newcommand{\bbL} {\mathbb{L}} \def\real{{\mathbb R}} \def\rational{{\mathbb Q}} \def\integer{{\mathbb Z}} \def\continuum{{\frak c}} \def\co{\continuum} \def\add{{\operatorname{A}}} \def\mul{{\operatorname M}} \def\ext{{\operatorname{Ext}}} \def\pr{{\operatorname{PR}}} \def\pc{{\operatorname{PC}}} \def\ac{{\operatorname{AC}}} \def\conn{{\operatorname{Conn}}} \def\darb{{\operatorname D}} \def\Ext{\ext} \def\pr{{\operatorname{PR}}} \def\PC{\pc} \def\AC{\ac} \def\Conn{\conn} \def\D{\darb} \def\proj{{\operatorname{pr}}} \def\supp{{\operatorname{supp}}} \def\cl{{\operatorname{cl}}} \def\bd{{\operatorname{bd}}} \def\cf{{\operatorname{cf}}} \def\diam{{\operatorname{diam}}} \def\rl{\real} \def\reals{\real} \def\mathR{\real} \def\ds{\bigcup} \def\di{\bigcap} \def\sb{\subseteq} \def\su{\subseteq} \newcommand{\propsub}{\subsetneqq} \def\sm{\setminus} \def\es{\emptyset} \def\B{{\cal B}} \def\C{{\cal C}} \def\F{{\cal F}} \def\G{{\cal G}} \def\H{{\cal H}} \def\J{{\cal J}} \def\T{{\cal T}} \def\e{\varepsilon} \def\la{\langle} \def\ra{\rangle} % characteristic function \newcommand{\charf}[1]{\mbox{\raise.48ex\hbox{$\chi$}$_{#1}$}} %% Theorems, etc. \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{problem}{Problem}[section] \newtheorem{example}{Example}[section] \newtheorem{definition}{Definition} \newtheorem{remark}{Remark}[section] \newcommand{\thm}[2]{\begin{theorem}\label{#1}#2\end{theorem}} \newcommand{\cor}[2]{\begin{corollary}\label{#1}#2\end{corollary}} \newcommand{\prop}[2]{\begin{proposition}\label{#1}#2\end{proposition}} \newcommand{\lem}[2]{\begin{lemma}\label{#1}#2\end{lemma}} \newcommand{\prob}[2]{\begin{problem}\label{#1}#2\end{problem}} \newcommand{\ex}[2]{\begin{example}\label{#1}#2\end{example}} \newcommand{\defi}[2]{\begin{definition}\label{#1}#2\end{definition}} \newcommand{\rem}[2]{\begin{remark}\label{#1}#2\end{remark}} \def\pf{\noindent{\sc Proof.} } \def\proof{\pf} %%%%%% \begin{document} \maketitle \begin{abstract} The main purpose of this paper is to describe two examples. The first is that of an almost continuous, Baire class two, non-extendable function $f\colon[0,1]\to[0,1]$ with a $G_\delta$ graph. This answers a question of Gibson~\cite{Gib}. The second example is that of a connectivity function $F\colon\real^2\to\real$ with dense graph such that $F^{-1}(0)$ is contained in a countable union of straight lines. This easily implies the existence of an extendable function $f\colon\real\to\real$ with dense graph such that $f^{-1}(0)$ is countable. We also give a sufficient condition for a Darboux function %from $f\colon[0,1]\to[0,1]$ with a $G_\delta$ graph whose closure is bilaterally dense in itself to be quasi-continuous and extendable. \end{abstract} \section{Definitions and Notation} Our terminology is standard and follows \cite{Ci:book}. We consider only real-valued functions of one or more real variables. No distinction is made between a function and its graph. A restriction of a function $f\colon X\to Y$ to a set $A\subset X$ is denoted by $f\restriction A$. %Symbol $\charf{A}$ will be used %for a characteristic function of a subset $A$ of a fixed space $X$. By $\real$ and $\rational$ we denote the set of all real and rational numbers, respectively, while $I$ will stand for the interval $[0,1]$. The closure of a set $A\subset\real^n$ is denoted by $\cl(A)$, its boundary by $\bd(A)$ and its diameter by $\diam(A)$. The first coordinate projection of a set $A\subset\real^2$ will be denoted by $\proj(A)$. The ordinal numbers will be identified with the sets of all their predecessors and cardinals with the initial ordinals. In particular $2=\{0,1\}$ and the first infinite ordinal $\omega$ number is equal to the set of all natural numbers $\{0,1,2,\ldots\}$. %The family of all functions from a %set $X$ into $Y$ is denoted by $Y^X$. In particular %$2^n$ will stand for the set of all sequences %$s\colon \{0,1,2,\ldots,n-1\}\to\{0,1\}$, while %$2^{<\omega}=\bigcup_{n<\omega}2^n$ is the set of all finite sequences %into $2$. %The symbol $|X|$ stands for the cardinality of a set $X$. The cardinality of %$\real$ is denoted by $\co$ and referred as {\em continuum}. %A set $S\subset\real$ is said to be {\em $\co$-dense\/} %if $|S\cap(a,b)|=\co$ for every $a1$, into $\real$ we have \hspace{-2.1pc} \begin{picture}(0,90) \put(120,55){\makebox(0,0){$\Ext=\Conn=\PC$}} \put(167,55){\vector(1,0){15}} \put(205,55){\makebox(0,0){$\AC\cap\D$}} \put(230,60){\vector(2,1){18}} \put(262,70){\makebox(0,0){${\AC}$}} \put(260,40){\makebox(0,0){${\D}$}} \put(230,52){\vector(2,-1){18}} \end{picture} where arrows denote strict inclusions. (Equation $\Ext=\Conn$ on $\real^n$ is proved in~\cite{CNW}.) \section{Almost Continuous Non-Extendable Function with $G_\delta$ Graph} As noted above the classes listed in (\ref{DLF}) cannot be distinguished within the family $\B_1$ of Baire one functions. On the other hand, the classes of functions on $\real$ or $I$ listed in (\ref{DLF}) can be distinguished within the family $\B_2$ of Baire class two functions. (See e.g. \cite[thm~1.2]{CiJa}.) The next natural question is: for which classes $\G$ containing $\B_1$ but not $\B_2$ will the inclusions remain strict? For several classes $\G$ strictly between $\B_1$ and $\B_2$ this has been investigated by Brown~\cite{Br} and Brown, Humke and Laczkovich~\cite{bhl}. In this section we will investigate the inclusions in (\ref{DLF}) within the class $\G=\G_\delta\cap\B_2$, where $\G_\delta$ is the class of functions with $G_\delta$ graphs. Notice that $\B_1\subset\G_\delta$ since for every pointwise limit $f$ of continuous functions $f_n\colon I\to\real$ the graph of $f$ is equal to $\bigcap_{n=1}^\infty \bigcup_{i=n}^\infty \{\la x,y\ra\colon |f_i(x)-y|<2^{-n}\}$. Clearly also every function $f$ from $\G_\delta$ is Borel, since for every open $U\subset\real$ the set $f^{-1}(U)=\proj(f\cap[I\times U])= I\setminus\proj(f\cap[I\times(\real\setminus U)])$ is simultaneously analytic and coanalytic, and hence Borel. (See e.g.~\cite[p.~489]{Kur} or \cite[p.~89]{Ke}.) On the other hand there are functions in $\G_\delta$ of arbitrary high Borel class. This follows from the fact that every Borel set is a one-to-one continuous image of a closed subset of $\real\setminus\rational$ (see e.g. \cite[15.3, p.~89]{Ke}) which is clearly $G_\delta$ in $\real$. Indeed, if $A\subset I$ is of the class at least $\alpha$ for some $\alpha<\omega_1$, take one-to-one continuous functions $f_0$ from a closed subset of $(0,1)\setminus\rational$ onto $A$ and $f_1$ from a closed subset of $(2,3)\setminus\rational$ onto $I\setminus A$. Then $f=f_0^{-1}\cup f_1^{-1}\colon I\to\real$ has a $G_\delta$ graph and is of Borel class at least $\alpha$, since $f^{-1}[(0,1)]=A$. Thus, if $\B_\alpha$ stands for the Baire class $\alpha$ functions, then we have the following relations for every $2\leq\alpha<\omega_1$, where arrows denote strict inclusions. \vspace{-15pt} \hspace{-2.1pc} \begin{picture}(0,100) \put(90,50){\makebox(0,0){$\B_1$}} \put(105,50){\vector(1,0){15}} \put(145,50){\makebox(0,0){$\G_\delta\cap \B_2$}} \put(169,55){\vector(2,1){18}} \put(200,65){\makebox(0,0){${\B_\alpha}$}} \put(210,64){\vector(2,-1){18}} \put(200,35){\makebox(0,0){${\G_\delta}$}} \put(169,47){\vector(2,-1){18}} \put(210,38){\vector(2,1){18}} \put(245,50){\makebox(0,0){$Borel$}} \end{picture} \vspace{-15pt} The properness of inclusions in (\ref{DLF}) within the class $\G_\delta$ has been summarized by Brown in~\cite{B}. (See also \cite[sec.~3]{GN}.) In particular, he noticed that the inclusions $\AC\subset\Conn\subset\D\subset\PC$ remain proper within this class $\G_\delta$ leaving open the problem of properness of inclusion $\Ext\subset\AC$ in the $\G_\delta$ class. (This problem is stated explicitly by Gibson in~\cite[Question~4]{Gib}.) In fact the examples exhibiting properness of the inclusions $\AC\subset\Conn\subset\D\subset\PC$ within the class $\G_\delta\cap\B_2$ can be found in the literature. For $\AC\subset\Conn$ Jones and Thomas~\cite{JT} (also see Thomas~\cite{Th}) constructed a non almost continuous function $f\colon I\to I$ with a connected $G_\delta$ graph. This $f$ is also $\B_2$, but it is not pointed out in the paper. Also Brown~\cite[remark~3]{Br} gives a bit stronger example and states explicitly that it is in $\G_\delta\cap\B_2\cap\Conn$ but not in $\AC$. The properness of the inclusion $\Conn\subset\D$ within $\G_\delta\cap\B_2$ is stated explicitly by Brown in~\cite[example~1 and remark~3]{Br}, using the example constructed by him in~\cite[example~2]{Br0}. (Also, in~\cite{Mil} Miller gives an example of a Darboux function $f\colon I\to I$ having a $G_\delta$ graph but no fixed point and so the graph of $f$ is not connected. This function is also $\B_2$, but it is not pointed out in the paper.) Concerning the properness of the inclusion $\D\subset\PC$ in $\G_\delta\cap\B_2$ Brown~\cite{B} states that there is one (not mentioning $\B_2$) and Miller~\cite{Mil} describes a peripherally continuous non Darboux function with $G_\delta$ graph, which turns out to be also in $\B_2$. The main goal of this section is to describe an example justifying properness of $\Ext\subset\AC$ within $\G_\delta\cap\B_2$, thus answering a question of Gibson in~\cite[Question~4]{Gib}. However, we will notice also that the small modifications of this example justify also properness of the remaining inclusions of (\ref{DLF}) within $\G_\delta\cap\B_2$. \begin{picture}(280,280)(-20,0) {\thicklines \put(0,10){\vector(1,0){260}} \put(0,10){\vector(0,1){260}} } \drawline[20](243,8)(243,12)\put(243,2){\makebox(0,0){{\small $1$}}} \drawline[20](-2,253)(2,253)\put(-8,253){\makebox(0,0){$1$}} \drawline[20](81,8)(81,12)\put(81,2){\makebox(0,0){$c_1$}} \drawline[20](162,8)(162,12)\put(162,2){\makebox(0,0){$d_1$}} \drawline[20](162,10)(81,131.5) \drawline[20](-2,131.5)(2,131.5)\put(-8,131.5){\makebox(0,0){$\frac{1}{2}$}} \drawline[20](27,8)(27,12)\put(27,2){\makebox(0,0){{\small $a_1$}}} \drawline[20](54,8)(54,12)\put(54,2){\makebox(0,0){{\small $b_1$}}} \drawline[20](27,10)(54,131.5) \drawline[20](189,8)(189,12)\put(189,2){\makebox(0,0){{\small $a_2$}}} \drawline[20](218,8)(218,12)\put(218,2){\makebox(0,0){{\small $b_2$}}} \drawline[20](189,10)(218,192.25) \drawline[20](-2,192.25)(2,192.25) \put(-8,192.25){\makebox(0,0){$\frac{3}{4}$}} \drawline[20](9,8)(9,12)\put(9,2){\makebox(0,0){{\small $c_2$}}} \drawline[20](18,8)(18,12)\put(18,2){\makebox(0,0){{\small $d_2$}}} \drawline[20](18,10)(9,192.25) \drawline[20](63,8)(63,12)\put(63,2){\makebox(0,0){{\small $c_3$}}} \drawline[20](72,8)(72,12)\put(72,2){\makebox(0,0){{\small $d_3$}}} \drawline[20](72,10)(63,222.625) \drawline[20](-2,222.625)(2,222.625) \put(-8,222.625){\makebox(0,0){$\frac{7}{8}$}} \drawline[20](171,8)(171,12)\put(171,2){\makebox(0,0){{\small $c_4$}}} \drawline[20](180,8)(180,12)\put(180,2){\makebox(0,0){{\small $d_4$}}} \drawline[20](180,10)(171,237.8125) \drawline[20](-2,237.8125)(2,237.8125) \put(-8,237.8125){\makebox(0,0){$\frac{15}{16}$}} \drawline[20](227,8)(227,12)\put(227,2){\makebox(0,0){{\small $c_5$}}} \drawline[20](236,8)(236,12)\put(236,2){\makebox(0,0){{\small $d_5$}}} \drawline[20](236,10)(227,245.40575) \drawline[20](-2,245.40575)(2,245.40575) \end{picture} \begin{center} Figure 1: function $f\in \G_\delta\cap\B_2\cap\AC\setminus\Ext$ from Example~\ref{ex:Gd} \end{center} \ex{ex:Gd}{There exists an almost continuous, non-extendable, Baire class two function $f\colon I\to I$ with $G_\delta$ graph.} %which is not extendable.} \proof The example is a slight modification of a function constructed by Ciesielski and Jastrz{\c{e}}bski in~\cite[thm~3.1]{CiJa}. It also slightly simplifies the argument given there. Let $C$ be the Cantor ternary set in $I$ and let ${\cal J}$ be the family of all component intervals of $I\setminus C$. For $i\in\{0,1\}$ let ${\cal J}_i$ be the family of all intervals $J\in{\cal J}$ of length $3^{-2n+i}$ for some integer $n$. Thus, $\{{\cal J}_0,{\cal J}_1\}$ is a partition of ${\cal J}$. Let $\{(a_n,b_n)\colon 00$. Then $p_n$ is the upper endpoint for $f$ of some interval $J_0\in\J$. We have to find $B_n$ and $p_{n+1}$. So, choose an $\e>0$ less than the length of $J_0$ %$\e\in(0,2^{-n+1})$ less than the length of $J_0$ %and $\diam(B_n)$ such that $\e<\min\{p_n,1-p_n\}$ and such that $\e<\min\{2^{-(n+1)},p_n,1-p_n,\diam(B_{n-1})\}$ and \begin{description} \item{($*$)} if $J\in\J\setminus\{J_0\}$ is closer to $p_n$ than $\e$, then it has the length at most $2^{-(n+1)}$ and $f(p)\geq 1-2^{-(n+1)}$ for the upper endpoint $p$ for $f$ of $J$. \end{description} Since $F(p_{n},0)\geq 1-2^{-n}$ and $F$ is peripherally continuous we can find an open neighborhood $W_n\subset[0,1]^2$ of $\la p_n,0\ra$ with diameter less than $\e$ and such that $F[\bd(W_n)]\subset(1-2^{-n+1},1]$. Without loss of generality we can also assume that $\bd(W_n)$ is connected. (This is a standard argument. See e.g.~\cite{JS}.) Note that the choice of $\e$ guarantees that $\bd(W_n)\cap B_{n-1}\neq\emptyset\neq\bd(W_n)\cap(I_0\times\{0\})$, where $I_0$ is a component of $I\setminus J_0$ containing $p_n$. This is so, since $\bd(W_n)$ disconnects $[0,1]^2$, while $B_{n-1}$ and $I_0\times\{0\}$ are connected, containing $\la p_n,0\ra\in W_n$ and of diameter greater than $\e\geq\diam(\bd(W_n))$. Let $z\in I_0$ be such that $\la z,0\ra\in \bd(W_n)\cap (I_0\times\{0\})$. Since $F(z,0)\in F[\bd(W_n)]\subset(1-2^{-n+1},1]\subset(0,1]$, there exists a $J\in\J$ such that $z\in\cl(J)$. Let $p_{n+1}$ be the upper endpoint for $f$ of $J$. Then, by ($*$), $f(p_{n+1})\geq 1-2^{-(n+1)}$ and the distance of $\la p_{n+1},0\ra$ from $\la z,0\ra\in \bd(W_n)$ is at most $2^{-(n+1)}$. The set $B_n$ is defined as a union of $\bd(W_n)$ and a closed segment joining $\la z,0\ra$ and $\la p_{n+1},0\ra$. It is easy to see that the inductive conditions are satisfied. This finishes the construction. Now, to finish the argument notice that for every $n<\omega$ the set $B^n=\bigcup_{k=n}^\infty B_k$ is connected, has diameter at most $\sum_{k=n}^\infty 2^{-k}=2^{-n+1}$ and contains $\la p_{n+1},0\ra$. In particular, there exists $p\in I$ with $p=\lim_n p_n$. We claim that $F(p,0)=1$. Indeed, if this is not the case, then there exists an $n<\omega$ such that $|F(p,0)-1|>2^{-n+2}$. Take $\e>0$ less than the diameter of $B^n$ and let $U$ be an open neighborhood of $\la p, 0\ra$ of diameter less than $\e$ and such that $F[\bd(U)]\subset(F(p,0)-2^{-n+1},F(p,0)+2^{-n+1})$. Then there exists a point $w\in B^n\cap\bd(U)$, since $U\cap B^n\neq\emptyset$, $U$ has diameter less than $\diam(B^n)$ and $B^n$ is connected. But then $F(w)$ belongs to both $(F(p,0)-2^{-n+1},F(p,0)+2^{-n+1})$ and $F[B^n]\subset(1-2^{-n+1},1]$, which is impossible, since these two sets are disjoint. So, $F(p,0)=1$. But this is impossible as well since then we would have $f(p)=F(p,0)=1$ and $f$ does not attain $1$. This contradiction finishes the proof that $f$ is not extendable. To show that $f$ is almost continuous let $G$ be an open subset of $I^2$ containing the graph of $f$. Notice that for every $x\in[0,1]$ there exists an interval $(r_x,s_x)$ (for $x=0$ and $x=1$ we consider intervals $[r_x,s_x)$ and $(r_x,s_x]$, respectively) such that \begin{itemize} \item $x\in(r_x, s_x)$, $f(r_x)=f(s_x)=0$ and \item there is a continuous function $g_x \colon [0,1] \to \mathR$ with $g_x \restriction{[r_x,s_x]} \subset G$ and such that $g_x(t)=0$ for $t \notin (r_x, s_x)$. \end{itemize} Indeed, if $f(x)=0$, then it is easy to find $r_x0$. Let $U\subset G$ be an open circular neighborhood of $\la b,f(b)\ra$. Then there exists an interval $J'=(c,d)\in\J_1$ with $bd_n>b_n$ from $(1-2^{-n},1)$, pick $a_n0$, $k\in\{-1,1\}$, then for every triangle $T$ from $S_{n-1}$ for which $(k\cdot f_n)[T]\subset[0,\infty)$ the following holds. \begin{description} \item{(a)} With every non-zero dyadic number $i/2^{n}\in[-2^n,2^n]$, where $i$ belongs to $D_n=\{i\in\integer\colon -4^{n}\leq i\leq 4^{n}\ \&\ i\neq 0\}$, we associate a triangle $T_i\su\hat{T}$ from $S_n$ such that all triangles $T_i$ are disjoint, $\hat f_n\left[\hat T\setminus\bigcup_{i\in D_n}T_i\right]=\{k\}$ and $\hat f_n\left[\hat T_i\right]=\{i/2^{n}\}$ for every $i\in D_n$. \item{(b)} For every $x\in T\setminus \hat T$ either $\hat f_{n-1}(x)\leq\hat f_{n}(x)\leq k$ or $\hat f_{n-1}(x)\geq\hat f_{n}(x)\geq k$. % $\hat f_{n-1}(x)$ and $\hat f_{n}(x)$ remain on the same side of $k$ % and $|\hat f_n(x)-k|\leq |\hat f_{n-1}(x)-k|$ % for every $x\in T\setminus \hat T$. \end{description} \end{description} To start the induction define grid $S_0$ as in Figure~3 with all sides of length $1$ and let $f_0\colon S_0\to\real$ be identically $1$. It is easy to see that conditions (i)--(iv) are satisfied for such a choice. Next, assume that for some $n>0$ we already have $S_{n-1}$ and $f_{n-1}$ satisfying (i)--(iv). We will define $S_n$ and extend $f_{n-1}$ to $f_n\colon S_n\to\real$ such that (i)--(iv) will still hold. Let $T$ be a triangle from $S_{n-1}$ and put $k=1$ if $\hat f_{n-1}[T]\subset [0,\infty)$ and $k=-1$ for $\hat f_{n-1}[T]\subset (-\infty,0]$. The $S_n$-triangulation of $T$ will be a refinement of the basic triangulation of $T$. This will guarantee (ii). Clearly $f_n$ is already defined on $\bd(T)$. We define $f_n$ on each vertex of the middle quarter $\hat{T}$ of $T$ by assigning it the value $k$. Next, on each triangle $T'$ from the basic partition of $T$ except for $\hat T$ (see Figure~4) we extend $f_n$ to $\hat f_n$ linearly. Such an extension is unique since $f_n$ is already defined on each vertex of $T'$. Note that this and the choice of $k$ guarantee (iv)(b). This also gives us (i) for any triangle from a subtriangulation of $T'$. \bigskip \begin{picture}(300,220)(-40,0) \thicklines \multiput(10,10)(60,0){4}{ \dashline{4}(0,0)(60,0)(30,52)(0,0)} \multiput(40,62)(60,0){3}{ \dashline{4}(0,0)(60,0)(30,52)(0,0)} \multiput(70,114)(60,0){2}{ \dashline{4}(0,0)(60,0)(30,52)(0,0)} \put(100,166){ \dashline{4}(0,0)(60,0)(30,52)(0,0)} \multiput(55,36)(60,0){3}{ \drawline[20](0,0)(30,0)(15,26)(0,0)} \multiput(85,88)(60,0){2}{ \drawline[20](0,0)(30,0)(15,26)(0,0)} \put(115,140){ \drawline[20](0,0)(30,0)(15,26)(0,0)} \end{picture} \begin{center} Figure 5: some triangles $T_i$ of the grid of $\hat{T}$ \end{center} Thus, $\hat f_n$ is defined so far on $T\setminus \hat T$. To extend $\hat f_n$ to $\hat T$ we proceed as follows. Partition $\hat{T}$ into a grid $S$ such that $S$ contains $2\cdot 4^n$ disjoint triangles $\{T_i\colon i\in D_n\}$. (See Figure~5.) We extend $\hat f_n$ to $\hat T\setminus\bigcup_{i\in D_n}T_i$ by assigning it the constant value $k$, while on each $\hat T_i$ with $i\in D_n$, $\hat f_n$ is identically $i/2^{n}$. This guarantees the reminder of condition (iv) as well as (i) for the triangles considered so far. Finally, $\hat f_n$ is extended linearly on each triangle from the basic partition of $T_i$ using the fact that it is already defined on each of its vertices. This finishes the construction of $\hat f_n$. Now, to make sure that (iii) and the remaining part of (i) are satisfied we will refine the triangulations defined so far to the grid $S_n$. This will lead to the definition of $f_n$ as $\hat f_n\restriction S_n$. For (i) we proceed as follows. If $k$ and $i\in D_n$ have the same sign, then $\hat f^n\restriction T_i$ does not attain value $0$ and (i) is already satisfied by the triangles from the basic partition of $T_i$. In this case $S_n$ on $T_i$ will be formed from the triangles considered so far. On the other hand, if $k$ and $i\in D_n$ have different signs, then $(\hat f^n\restriction T_i)^{-1}(0)$ is a boundary of a triangle $T'$ similar to $T_i$. Then we subtriangulate the basic partition of $T_i$ in such a way that every edge of $T'$ is covered by the edges of this subtriangulation and that each triangle from the subtriangulation intersects at most one edge (excluding the end points) of $T'$. If, in addition, $(\hat f^n\restriction T_i)^{-1}(-k)$ is nonempty, then it is a boundary of a triangle (or the entire triangle) and we will make sure that this boundary is a subset of our subtriangulation. Condition (iii) is easily guaranteed by refining the triangulation described so far. This finishes the inductive construction. \medskip Now, $F$ is defined on $S=\ds_{n<\omega}S_{n}$ as $f=\ds_{n<\omega}f_{n}$. We will extend it to $\real^2$ making sure that $F(x)\neq 0$ for every $x\in\real^{2}\sm S$. This will guarantee that $F^{-1}(0)\subset S$. Note also that, by (iv)(a), the graph of $f$ is already dense in $\real^3$. So any extension $F$ of $f$ will have this property as well. To extend $F$ to $\real^2$ pick an $x\in\real^{2}\sm S$. Note that, by (i), $\hat f_n(x)\neq 0$ for every $n<\omega$. For every $n<\omega$ let $T^x_n$ be the triangle from $S_n$ containing $x$ and let $N=\{n<\omega\colon x\in\hat T^x_n\}$. We consider two cases. \smallskip \noindent Case 1. The set $N$ is infinite. If $N^k=\{n\in N\colon \hat f_n[\hat T^x_n]=\{k\}\}$ for $k\in\{-1,1\}$, then $N=N^{-1}\cup N^1$ and so at least one of the sets $N^{-1}$ and $N^1$ is infinite. Let $k=1$ if $N^1$ is infinite and put $k=-1$ otherwise. In this case we define $F(x)=k$. Notice that this guarantees that $F$ is peripherally continuous at $x$ since for any $n\in N^k$ \[ F[\bd(\hat T^x_n)]=f_n[\bd(\hat T^x_n)]=\{k\}=\{F(x)\}, \] $x$ belongs to the interior of $\hat T^x_n$ and the diameter of $\hat T^x_n$ is at most $1/2^n$. \smallskip \noindent Case 2. The set $N$ is finite. Fix an $m<\omega$ such that $N\subset\{0,\ldots,m-2\}$. Then $x\in T^x_n\setminus \hat T^x_n$ for every $n\geq m$. So, if $k\in\{-1,1\}$ is such that $(k\cdot f_m)[T^x_m]\subset[0,\infty)$, then, by (iv)(b), either $\hat f_{m}(x)\leq\hat f_{m+1}(x)\leq\hat f_{m+2}(x)\leq\cdots\leq k$ or $\hat f_{m}(x)\geq\hat f_{m+1}(x)\geq\hat f_{m+2}(x)\geq\cdots\geq k$. So the limit $L=\lim_{m\to\infty}\hat f_{m}(x)$ is well defined and not equal to $0$. We put $F(x)=L$. This, together with (ii) and (iii), guarantees that $F$ is peripherally continuous at $x$. This finishes the construction of function $F$ and the argument that $F$ is peripherally continuous on $\real^2\setminus S$. To see that $F$ is peripherally continuous on $S$ take an $x\in S$. Then, there exists a $k<\omega$ such that $x \in S_{n}$ for every $n\geq k$. For any such $n$ let $\T_n$ be the set of all triangles from $S_n$ to which $x$ belongs. Then $x$ belongs to the interior of the polygon $P_n=\bigcup\T_n$. Moreover, by (ii) and (iii), the variation on the boundary of $P_n$ and the diameter of $P_n$ are at most $2/2^n$. So, the sequence $\la P_n\ra$ guarantees that $F$ is peripherally continuous at $x$. This finishes the proof of Theorem~\ref{thConnExamp}. \qed \section{Quasi-Continuous Extendable Functions} To state the results of this section we need the following additional terminology and facts. For $x\in I$ let $l_x=\{x\}\times I$ and for function $f\colon I\to I$ let $C(f)$ stand for the set of points of continuity of $f$. Recall that for a Darboux function $f\colon I\to I$ the set $\cl(f)\cap l_x$ is connected for every $x\in I$ and that $C(f)$ is a dense $G_\delta$ provided $f$ has a $G_\delta$ graph. (See e.g.~\cite{JT}.) The function $f\colon I\to I$ is {\em quasi-continuous\/} if $f\restriction C(f)$ is dense in the graph of $f$. A function $f\colon I\to I$ is said to have a closure that is {\em bilaterally dense in itself\/} if $\cl(f\restriction(0,x))\cap l_x=\cl(f\restriction(x,1))\cap l_x$ for each $x\in(0,1)$. Croft's function $f\colon I\to I$ from~\cite{Croft} is Darboux, upper semicontinuous (hence of Baire class one) and $0$ almost everywhere, but is not identically $0$. It follows that the closure of $f$ is bilaterally dense in itself. However, since it is not quasi-continuous, Croft's function does not satisfy the second category condition in the following result. But a function like Kellum and Garrett's~\cite[Example~1]{KG} does satisfy it and the rest of the hypothesis. \thm{thS4}{ Suppose $f\colon I\to I$ is a Darboux function with a $G_\delta$ graph whose closure is bilaterally dense in itself. Also suppose for every point $\la x,f(x)\ra$ of $f$ there exists an open neighborhood $W\subset I^2$ of $\la x,f(x)\ra$ such that if $B$ is closed and nowhere dense in $\proj(f\cap W)$, then $\proj^{-1}(B)\cap (f\cap W)$ is nowhere dense in $f\cap W$. Then $f$ is quasi-continuous and extendable. } \proof Let $A=\{x\in I\colon \cl(f)\cap l_x\mbox{ is nondegenerate}\}$. Assume $f$ is not quasi-continuous at some point $x_0\in I$. Then there exists a rectangular open neighborhood $W=I_1\times I_2\subset I^2$ of $\la x_0,f(x_0)\ra$ that obeys the second category condition of the hypothesis and that contains no point of $f\restriction C(f)$. Note that $\proj(f\cap W)=\{x\in A\cap I_1\colon \la x,f(x)\ra\in W\}$. According to the Alexandroff theorem, the $G_\delta$ subset $f\cap W$ of $I^2$ is homeomorphic to a complete metric space. Therefore, by hypothesis, $\proj(f\cap W)$ is of second category in itself. Observe that if $x\in\proj(f\cap W)$, then $\cl(f)\cap l_x$ is a nondegenerate interval meeting $I^2\setminus W$ and so $\cl(f\cap W)\cap l_x$ contains a nondegenerate interval containing $\la x,f(x)\ra$. To show $f\cap W$ is nowhere dense in $\proj(f\cap W)\times I$ let $[a,b]\times[c,d]$ be a rectangle in $I^2$ that meets $\proj(f\cap W)\times I$ and let $G_1\supset G_2\supset\cdots$ be open subsets of $I^2$ such that $f\cap W=\left(\bigcap_{n=1}^\infty G_n\right)\cap(\proj(f\cap W)\times I)$. For all positive integers $n$ and rational numbers $r$ and $s$ with $c\leq r