The goal of this paper is to characterize the functions that can be written as the product of bounded quasi-continuous functions.

Let $f$ be an interval map in a neighborhood of the fixed point $0$ with \(-1<\lambda=f'(0)<0\). Continuity of $f$ is not assumed at points other than the fixed point. It is shown that if either $$f\circ f(x)\geq\lambda^{2}x \text{ or }f\circ f(x)\leq\lambda^{2}x$$ for each $x$ in a neighborhood of $0$, then the Koenigs' sequence \(\{\phi_{k}\}\) defined by $\phi_{k}(x)=\dfrac{f^{k}(x)}{\lambda^{k}}$ converges uniformly to a limit \(\phi\) in a neighborhood of $0$ with \(\phi(0)=0\) and \(\phi'(0)=1\). Two examples are presented, the first of which is a \(C^{1}\) map $f$ with \(f(0)=0\) and \(-1<f'(0)<0\) having a divergent Koenigs' sequence. The other example is a \(C^{1}\) convex map $g$ with \(g(0)=0\) and \(0<g'(0)<1\) for which the associated Koenigs' sequence diverges and which has no orientation-reversing composition square root that is differentiable at $0$.

Given a set $M\subset\R$ of Lebesgue measure zero, let $\B_1\vert_M$ be the set of all restrictions to $M$ of bounded Baire one functions on $\R$ and $\A$ the set of all bounded approximately continuous functions on $\R$. We discuss the existence of simultaneous extension operators for $\B_1\vert_M$ and $\A$. We show that there exists a~positive linear operator $L\colon\B_1\vert_M\to\A$ such that $L(g)\vert_M=g$ for all $g\in\B_1\vert_M$, if and only if $M$ is a~scattered set and this is the case if and only if there exists a~continuous linear operator $L_1\colon\B_1\vert_M\to\A$ with the same property. Also, we show that there exist non-regular continuous linear operators $T_1\colon \ell_\infty \to \A$ and $T_2\colon\A\to\A$.

: 9/8/99 cew % last edit: 9/8/99 cew % gallies sent: 9/21/99 % gallies corrected: cew 10/6/99 % set in production style: 9/8/99 cew % Research Section from Editor Thomson \documentclass{rae} \usepackage{amsmath,amsthm,amssymb} %\coverauthor{Zbigniew Grande} %\covertitle{On A. C. Limits of Decreasing Sequences of Continuous or Right Continuous Functions} \received{August 3, 1998} \MathReviews{26A15, 26A21, 26A99} \keywords{upper semicontinuity, decreasing sequences of functions, $B_1^*$ class, right continuity.} \firstpagenumber{231} \markboth{Zbigniew Grande}{On A. C. Limits of Decreasing Sequences} \author{Zbigniew Grande\thanks {Supported by Bydgoszcz Pedagogical University grant 1998} , Institute of Mathematics, Pedagogical University, Plac Weyssenhoffa 11, 85-072 Bydgoszcz, Poland. e-mail: {\tt grande@wsp.bydgoszcz.pl}} \title{ON A. C. LIMITS OF DECREASING SEQUENCES OF CONTINUOUS OR RIGHT CONTINUOUS FUNCTIONS} %%%Put Autnor's Definitons Below Here%%% %\newcommand{\mathR}{\mathop{\rm I\kern-0.35ex R}\nolimits} \newtheorem {thm} {Theorem} \newtheorem {lem} {Lemma} \newtheorem {rem} {Remark} {\theoremstyle{definition} \newtheorem{example}{Example}} \DeclareMathOperator{\cl}{cl} \DeclareMathOperator{\a.c.}{a.c.} %%%%%% \newcommand{\rest}{\restriction} \DeclareMathOperator{\osc}{osc} \newcommand{\R}{\ensuremath{\mathbb R}} \newcommand{\pf}{\noindent{\sc Proof.~}} \begin{document} \maketitle \begin {abstract} The a.c. limits ({\bf i.e.} the discrete limits introduced by Cs{\'a}sz{\'a}r and Laczkovich) of decreasing sequences of continuous (resp. right continuous) functions are investigated. \end {abstract} Let ${\R}$ be the set of all reals. $(X,\tau )$ or $X$ in this paper always denotes a perfectly normal Hausdorff topological space. A function $f:X \to {\R}$ is a $B_1^*$ function (belongs to the class $B_1^*$) if there is a sequence of continuous functions $f_n:X \to {\R}$ with $f = \a.c.\lim_{n \rightarrow \infty }f_n$, {\bf i.e.} for each point $x \in X$ there is a positive integer $k$ such that $f_n(x) = f(x)$ for every $n > k$ (compare \cite{2, 3}). From the results obtained in \cite{2} it follows that the function $f:X \to {\R}$ belongs to ${\cal B}_1^*$ if and only if there are closed sets $A_n$, $n = 1,2,\ldots $, such that the restricted functions $f\restriction A_n$ are continuous and $X = \bigcup_{n=1}^{\infty }A_n.$ \section{The Discrete Limits of Decreasing Sequences of Continuous Functions.} In the first part of this article we will investigate $B_1^*$ functions which are upper semicontinuous. Recall that the function $f:X \to {\R}$ is upper semicontinuous if for every real $a$ the set $\{ x \in X;f(x) < a\} $ belongs to $\tau $. Evidently the pointwise limit of each decreasing sequence of upper semicontinuous functions $f_n:X \to {\R}$, $n = 1,2,\ldots $, is upper semicontinuous. The following theorem can be found on page 51 of \cite{R}. \begin{rem} If the function $f:X \to {\R}$ is upper semicontinuous, then there is a decreasing sequence of continuous functions $f_n:X \to {\R}$, $n = 1,2,\ldots $, such that $f = \lim_{n \rightarrow \infty }f_n$. \end{rem} %%% We will prove the following theorem. \begin{thm} Let $(X,\tau )$ be a perfectly normal $\sigma $-compact Hausdorff topological space. Then an upper semicontinuous function $f:X \to {\R}$ belongs to class $B_1^*$ if and only if there is a decreasing sequence of continuous functions $f_n:X \to {\R}$ such that $f = \a.c.\lim_{n \rightarrow \infty }f_n$. \end{thm} We start from the following lemma. \begin{lem} Let $f:X \to {\R}$ be a function. If there are sets $A_n$ and continuous functions $f_n:X \to {\R}$ such that $A_1 \subset A_2 \subset \cdots$, $X = \bigcup_nA_n,$ $f_n \geq f$ and $f_n\rest A_n = f\rest A_n$ for $n = 1,2,\ldots,$ then there is a decreasing sequence of continuous functions $g_n:X \to {\R}$ with $f = \a.c.\lim_{n \rightarrow \infty }g_n$. \end{lem} \pf Of course, the functions $g_n = \min_{k \leq n}f_n$ satisfy all required conditions.\qed \medskip {\noindent{\sc Proof of Theorem 1.~} If $f$ is the discrete limit of a decreasing sequence of continuous functions $f_n:X \to {\R}$, then evidently $f \in {\cal B}_1^{*}$. So, we assume that $f \in {\cal B}_1^*$. Since $f$ is upper semicontinuous and $X$ is perfectly normal, by Remark 1 there is a decreasing sequence of continuous functions $f_n:X \to {\R}$ which converges to $f$ at each point $x \in X$. On the other hand $f$ is the discrete limit of continuous functions; so there are closed sets $A_n$, $n = 1,2,\ldots $, such that every restricted function $f\rest A_n$ is continuous and $X = \bigcup_{n=1}^{\infty }A_n.$ We can assume that $A_n$ is compact for each $n = 1,2,\ldots $. Fix a positive integer $k$. On $A_k$ the sequence $(f_n)$ tends uniformly to $f$ due to Dini's lemma. So we can also assume that $$\max\{ (f_n(x) - f_{n+1}(x));x \in A_k\} \leq 2^{-n}.$$ By Tietze's theorem for $n = 1,2,\ldots $ there is a continuous extension $g_n:X \to [0,2^{-n}]$ of the restricted function $(f_n - f_{n+1})\rest A_k$. Let $$h_n = \min(g_n,f_n - f_{n+1})\text{ for }n = 1,2,\ldots ,$$ and let $l_k = f_1 - \sum_{n=1}^{\infty }h_n.$ Since the series $\sum_{n=1}^{\infty }h_n$ converges uniformly, the function $l_k$ is continuous. Moreover, for $k = 1,2,\ldots $ we have $l_k \geq f$ and $f\rest A_k = l_k\rest A_k.$ So, by Lemma 1 we obtain our theorem. \qed \bigskip Theorem $1$ in the presented form and its proof was proposed by the referee. My formulation concerned the function $f:[a,b] \to {\R}$ and the Euclidean topology and its proof was more complicated. \section{Decreasing Sequences of Right Continuous Functions} In this part we assume that $X = [a,b)$ and $\tau $ is the topology of right continuity. This topology $\tau $ is perfectly normal and Hausdorff but is not $\sigma $-compact. So, the limit $f$ of a decreasing sequence of right upper semicontinuous functions $f_n$, $n = 1,2,\ldots $, is a right upper semicontinuous function and Remark 1 is valid for $(X,\tau )$. Thus we have the following assertion. \begin{rem} For every right upper semicontinuous function $f$ there is a decreasing sequence of right continuous functions $f_n$, $n = 1,2,\ldots $, such that $f = \lim_{n \rightarrow \infty }f_n$. \end{rem} From the last remark by an elementary proof we obtain the next assertion. \begin{rem} If a function $f:[a,b) \to {\R}$ is right upper semicontinuous, then there is a decreasing sequence of functions $f_n:[a,b) \to {\R}$ such that \begin{itemize} \item[] the functions $f_n$ are right continuous ; \item[] $f = \lim_{n \rightarrow \infty }f_n$; \item[] all functions $f_n$, $n = 1,2,\ldots $, are locally constant from the right, {\bf i.e.} for each point $x \in [a,b)$ there is a positive real $r_{x,n}$ such that $$I_{x,n} = [x,x+r_{x,n}] \subset [a,b)\text{ and }f\rest I_{x,n} \text{ is constant };$$ \item[] if $\limsup_{t \rightarrow x+}f(t) < f(x)$, then for $n$ sufficiently large $f_n(x) = f(x)$; \item[] for every integer $n$ the inclusion $f_n([a,b)) \subset \cl(f[a,b))), $ (where $\cl$ denotes the closure operation) holds. \end{itemize} \end{rem} \pf The set $A$ of all points $x$ at which $\limsup_{t \rightarrow x+}f(t) < f(x)$ is countable, i.e. if $A \neq \emptyset,$ then $A = \{ x_1,x_2,\ldots \} $. By Remark 2 there is a decreasing sequence of right continuous functions $g_n$ such that $f = \lim_{n \rightarrow \infty }g_n$. Fix a positive integer $n$ and observe that there is a sequence of intervals $I_{i,n} = [u_{i,n},v_{i,n})$, $i = 1,2,\ldots $, such that: \begin{itemize} \item[] $[a,b) = \bigcup_iI_{i,n}$; \item[] $I_{i,n} \cap I_{j,n} = \emptyset $ for $i \neq j$; \item[] $u_{i,n} = x_i$ for $i \leq n$; \item[] $\osc g_n < \frac{1}{n}$ on each interval $I_{i,n}$; \item[] $g_n(x) > f(x)$ if $x \in I_{i,n}$ and $i \leq n$. \end{itemize} Let $$h_n(x) =\begin{cases} f(x_i)&\text{for }x \in I_{i,n},\;\;i \leq n\\ \sup_{I_{i,n}}g_n&\text{for } x \in I_{i,n}\;\;i > n.\end{cases}$$ Then the functions $f_n = \min(h_1,h_2,\ldots ,h_n), $ for $n = 1,2,\ldots ,$ satisfy all required conditions.\qed \begin{thm} If $f = \a.c.\lim_{n \rightarrow \infty }f_n$, where all functions $f_n$, $n = 1,2,\ldots $, are right continuous, then $f$ satisfies the following condition. \begin{itemize} \item[(1)] For each nonempty perfect set $A \subset [a,b)$ there is an open interval $I$ such that $A \cap I \neq \emptyset $ and the restricted function $f\rest(I \cap B)$, where $$B = \{ x \in A;x\text{ is a right limit point of } A\} ,$$ is right continuous at each point of the intersection $B \cap I$. \end{itemize} \end{thm} \pf Let $A \subset [a,b)$ be a nonempty perfect set and let $B$ denote the set of all right limit points of $A$. For each point $x \in [a,b)$ there is a positive integer $n(x)$ such that $f_n(x) = f(x)$ for $ n \geq n(x).$ For $n = 1,2,\ldots $ put $A_n = \{ x \in [a,b);n(x) = n\} $ and observe that $[a,b) = \bigcup_{n=1}^{\infty }A_n.$ So there are an open interval $I$ and a positive integer $k$ such that $I \cap B \neq \emptyset$ and $A_k \cap I \cap B$ is dense in $B \cap I.$ Thus $f(x) = f_k(x)$ for each point $x \in I \cap A$ which is a right limit point of $A$ and consequently $f\rest(B \cap I)$ is right continuous at each point of $I \cap B$. \qed \medskip The above proof of Theorem $2$ is short. However the referee related this statement to the result of Cs{\'a}sz{\'a}r and Laczkovich (Theorem $13$ of \cite{2}, pp. 469) which says that if $X$ is a Baire space, the functions $f_n:X \to {\R}$, $n = 1,2,\ldots $, are continuous and $f = \a.c.\lim_{n \rightarrow \infty }f_n$, then the points of discontinuity of $f$ constitute a nowhere dense set in $X$. \medskip The connection between these two results is the following assertion. \medskip Let $(X,{\cal T},\tau )$ be a bitopological space such that $\tau $ is finer than ${\cal T}$ and $(X,{\cal T})$ is a Baire space. Assume that for every nonempty set $A \in \tau $ there is a nonempty set $B \in {\cal T}$ such that $B \subset A$. Then every $\tau $-nowhere dense set is ${\cal T}$-nowhere dense and $(X,{\cal T})$ is a Baire space. \medskip We arrive at Theorem $2$ at once if we observe that the sets $X \subset [a,b)$ having no right isolated points satisfy the conditions of the previous statement. So the quoted theorem of Cs{\'a}sz{\'a}r and Laczkovich can be applied. \medskip \begin{example}\label{ex1} Let $C$ be the Cantor ternary set and let $I_n = (a_n,b_n)$, $n = 1,2,\ldots $, be an enumeration of all components of the set $[0,1) \setminus C$ such that $I_n \cap I_m = \emptyset$ for $n \neq m,\;n,\,m = 1,2,\ldots .$ Put \begin{displaymath} f(x) = \left\{ \begin{array}{ccl} 1 & \text{for}& x \in B = C \setminus \{ a_n;n \geq 1\} \\ 0 &\text{for} & x \in [0,1) \setminus B. \end{array} \right. \end{displaymath} Observe that the function $f$ is not of Baire class one. For $n \geq 1$ let \begin{displaymath} f_n(x) = \left\{ \begin{array}{ccl} 0 &\text{for} & x \in B_n = \bigcup_{i \leq n}[a_i,b_i) \\ 1 &\text{for}& x \in [0,1) \setminus B_n. \end{array} \right. \end{displaymath} Then all functions $f_n$, $n = 1,2,\ldots $, are right continuous, $f_n \geq f_{n+1}$ for $ n = 1,2,\ldots $ and $\a.c.\lim_{n \rightarrow \infty }f_n = f.$ \end{example} Now we introduce the following condition $(1')$. \begin{itemize} \item[(1')] A function $f$ satisfies condition $(1')$ if for every nonempty closed set $A \subset [0,1)$ there is an open interval $I$ such that $I \cap A \neq \emptyset $ and the restricted function $f\rest(A \cap I)$ is right continuous. (If $x \in A$ is right isolated in $A$, then $f\rest A$ is right continuous at $x$ by default.) \end{itemize} Observe that the implication $(1') \Longrightarrow (1)$ is true. The function $f$ from Example $1$ satisfies condition $(1)$ but it does not satisfy condition $(1')$. Observe also that, by Baire's theorem on Baire $1$ functions, every function $f$ satisfying condition $(1')$ is of Baire $1$ class. \begin{thm} A function $f$ satisfies condition $(1')$ if and only if it satisfies the following condition. \begin{itemize} \item[(2)] There is a sequence of nonempty closed sets $A_n \subset [a,b)$ such that all restricted functions $f\rest A_n$, $n = 1,2,\ldots $, are right continuous and $[a,b) = \bigcup_{n=1}^{\infty }A_n.$ \end{itemize} \end{thm} \pf $(1') \Longrightarrow (2)$. We will apply transfinite induction. Let $I_0$ be an open interval with rational endpoints such that the restricted function $f\rest I_0$ is right continuous. Fix an ordinal number $\alpha > 0$ and suppose that for every ordinal number $\beta < \alpha $ there is an open interval with rational endpoints $I_{\beta }$ such that $H_{\beta } = I_{\beta } \setminus \bigcup_{\gamma < \beta }I_{\gamma } \neq \emptyset$ and the restricted function $f\rest H_{\beta }$ is right continuous. If $G_{\alpha } = [a,b) \setminus \bigcup_{\beta < \alpha }I_{\beta } \neq \emptyset,$ then by $(1')$ there is an open interval $I_{\alpha }$ with rational endpoints such that $I_{\alpha } \cap G_{\alpha } \neq \emptyset$ and the restricted function $f\rest (I_{\alpha } \cap G_{\alpha })$ is right continuous. Let $\xi $ be the first ordinal number $\alpha $ such that $[a,b) \setminus \bigcup_{\beta < \xi }I_{\beta } = \emptyset .$ Since the family of all intervals with rational endpoints is countable, $\xi $ is a countable ordinal number. Every set $H_{\alpha }$, $\alpha < \xi $, is an $F_{\sigma }$ set; so there are closed sets $H_{k,\alpha }$, $k = 1,2,\ldots $, such that $H_{\alpha } = \bigcup_{k=1}^{\infty }H_{k,\alpha } .$ Evidently, all restricted functions $f\rest H_{k,\alpha },\;k = 1,2,\ldots \text{ and } \alpha < \xi ,$ are right continuous. Now enumerate in a sequence $(A_n)$ all sets $$ H_{k,\alpha },\;\;k = 1,2,\ldots \;\; {\rm and} \;\; \alpha < \xi ,$$ and observe that this sequence satisfies all requirements. $(2) \Longrightarrow (1')$ Fix a nonempty closed set $A \subset [a,b)$. If $A$ contains isolated points, then condition $(1')$ is satisfied. So we assume that $A$ is a perfect set. By $(2)$ there is a sequence of closed sets $A_n$, $n = 1,2,\ldots $, such that $[a,b) = \bigcup_nA_n$ and all restricted functions $f\rest A_n$, $n = 1,2,\ldots $, are right continuous. Since $A = \bigcup_{n=1}^{\infty }(A \cap A_n),$ there are a positive integer $k$ and an open interval $I$ such that $I \cap A = I \cap A_k \neq \emptyset .$ But the restricted function $f\rest(A \cap I)$ is right continuous; so condition $(1')$ is satisfied.\qed \begin{thm} If $f$ satisfies condition $(1')$ (or equivalently $(2)$) from the last theorem, then there is a sequence of functions $f_n$, $n = 1,2,\ldots $, which are right continuous and for which $\a.c.\lim_{n \rightarrow \infty }f_n = f$. \end{thm} \pf There is a sequence of nonempty closed sets $A_n$, $n = 1,2,\ldots $, such that $[a,b) = \bigcup_{n=1}^{\infty }A_n,\; A_1 \subset A_2 \subset \cdots$ and all restricted functions $f\rest A_n$, $n = 1,2,\ldots $, are right continuous. By Tietze's theorem for $n = 1,2,\ldots $ there is a right continuous function $f_n:[a,b) \to {\R}$ which is equal to $f$ on the set $A_n$. Then $\a.c.\lim_{n \rightarrow \infty }f_n = f.$\qed \begin{thm} If a function $f$ is upper semicontinuous from the right and satisfies condition $(1')$ (or equivalently $(2)$), then there is a decreasing sequence of right continuous functions $f_n$, $n = 1,2,\ldots $, such that $\a.c.\lim_{n \rightarrow \infty }f_n = f.$ \end{thm} \pf Let a function $f$ satisfies the hypothesis of our theorem. There is a sequence of nonempty closed sets $A_n$, $n = 1,2,\ldots $, such that $[a,b) = \bigcup_{n=1}^{\infty }A_n$ and all restricted functions $f\rest A_n$, $n = 1,2,\ldots $, are right continuous. Without loss of the generality we can suppose that $a \in A_1 \subset A_2 \subset \ldots \subset A_n \subset \ldots .$ Fix a positive integer $n$ and enumerate in a sequence $(I_{n,k})_k$ all components of the set $[a,b) \setminus A_n$. If $I_{n,k} = (a_{n,k},b_{n,k})$, $ a_{n,k} \in A_n$ and $\limsup_{t \rightarrow a_{n,k}+}f(t) = f(a_{n,k})$, then we find points $c_{n,k,i}$, $i = 1,2,\ldots $, such that $$b_{n,k} = c_{n,k,1} > \ldots > c_{n,k,i} > \ldots \searrow a_{n,k}.$$ By Theorem 2 and Remark 1 for every $i \geq 1$ there is right constant function $h_{n,k,i}:[c_{n,k,i+1},c_{n,k,i}) \to {\R}$ such that $h_{n,k,i} \geq f/[c_{n,k,i+1},c_{n,k,i})$ and $h_{n,k,i}(c_{n,k,i}) < f(c_{n,k,i}) + \frac{1}{ik}.$ Let $g_{n,k,i}(x) = \max(h_{n,k,i}(x),f(a_{n,k}))$ for $x \in [c_{n,k,i+1},c_{n,k,i}).$ Observe that $g_{n,k,i}([c_{n,k,i+1},c_{n,k,i})) \subset \cl(f([a_{n,k},b_{n,k}))), \; i = 1,2,\ldots$ Next in every such interval $I_{n,k}$ we define the function $g_{n,k}$ by $$g_{n,k}(x) = g_{n,k,i}(x)\;\; {\rm for}\;\; x \in [c_{n,k,i+1},c_{n,k,i}), \;\;i = 1,2,\ldots .$$ If $I_{n,k} = [a_{n,k},b_{n,k}),\; a_{n,k} \in A_n,$ and $\limsup_{x \rightarrow a_{n,k}+}f(x) < f(a_{n,k}),$ then by Remarks $2$ and $3$ there is a right constant function $h_{n,k}:[a_{n,k},b_{n,k}) \to {\R}$ such that $h_{n,k} \geq f\rest [a_{n,k},b_{n,k})$, $ h_{n,k}(a_{n,k}) = f(a_{n,k})$ and $h_{n,k}([a_{n,k},b_{n,k})) \subset \cl(f([a_{n,k},b_{n,k})))).$ Let $g_{n,k}(x) = \max(h_{n,k}(x),f(x))$ for $ x \in [a_{n,k}, b_{n,k}).$ Put \begin{displaymath} g_n(x) = \left\{ \begin{array}{ccl} f(x) &\text{for} & x \in A_n\\ g_{n,k}(x) &\text{for} & x \in I_{n,k}\;\;i,k = 1,2,\ldots . \end{array} \right. \end{displaymath} Then the function $g_n$ is right continuous, $g_n \geq f$ and $g_n\rest A_n = f\rest A_n.$ So, by Lemma 1 we obtain our theorem.\qed \medskip Observe that the last theorem is not a corollary of Theorem $1$, since the topology $\tau $ is not $\sigma $-compact. \medskip \noindent{\bf Acknowledgment} I would like to thank to the referee for several suggestions, the reformulation of Theorem 1 and its proof. \noindent{\bf Problem.} Is Theorem 5 true if we replace condition $(1')$ by $(1)$? \begin{thebibliography}{9} \bibitem{1} A.~M.~Bruckner, J.~B.~Bruckner and B.~S.~Thomson, {\it Real Analysis}, Prentice--Hall International, INC, Upper Saddle River, New Jersey, 1996. \bibitem{2} A. Cs{\'a}sz{\'a}r and M. Laczkovich, {\it Discrete and equal convergence}, Studia Sci. Math. Hungar. 10 (1975), 463--472. \bibitem{3} R. J. O'Malley, {\it Approximately differentiable functions. The r topology}, Pacific J. Math. 72 (1977), 207--222. \bibitem{R}H. L. Royden, {\it Real Analysis\/}, Third Edition, MacMillan, 1988. %\bibitem{4} Sikorski R.;{\it Real Functions} Vol.I., (in Polish), Warsaw %PWN 1958. \bibitem{5} B. S. Thomson, {\it Real Functions}, Lectures Notes in Math. 1170 (1985), Springer--Verlag. \end{thebibliography} \par\newpage$\quad$\end{document} This remark is probably known, but I can not give any reference. So, I give an elementary proof. \bigskip By Th. $3.6$ from Sikorski's book \cite{4} (pp. 178) it suffices to prove that the characteristic function $\kappa _A$ of arbitrary set $A \in \tau $ is the limit of an increasing sequence of continuous functions. Fix a nonempty set $A \in \tau $. Since $X$ is perfectly normal, there are $\tau $-closed sets $A_1 \subset \ldots \subset A_n \subset \ldots $ such that $$A = \bigcup_{n=1}^{\infty }A_n .$$ By Tietze's theorem there are continuous functions $$g_n:X \to [0,1], \;\; n = 1,2,\ldots ,$$ such that $$g_n(A_n) = \{ 1\} \;\;\;{\rm and}\;\;\;g_n(X \setminus A) = \{ 0\} \;\;{\rm for}\;\;n = 1,2,\ldots .$$ Then the sequence of continuous functons $$f_n = \max (g_1,\ldots ,g_n) \;\;\; n = 1,2,\ldots ,$$ is creasing and converges to $\kappa _

Let $A$ and $B$ be sets of real sequences. Let $F(A,B)$ denote the set of functions $f : \R \rightarrow \R$ that preserve $A$ and $B$ in the sense that $(f(a_{n})) \in B$ for all sequences $(a_{n}) \in A.$ These functions are generalizations of convergence preserving functions first introduced by Rado. We establish identities and inclusions for $F(A,B)$ when $A$ and $B$ are $l^{p}$-spaces and other well-known sequence spaces. We also characterize $F(A,B)$ in terms of elementary classes of functions. Our characterizations are motivated by the work of Bors\'{\i}k, \v{C}erve\v{n}ansk\'{y} and \v{S}al\'{a}t.

A real-valued function $f$ defined on a topological space is called {\is absolutely polynomial} if its absolute value can be written as a polynomial in $f$ with continuous coefficients. One motivation for studying such functions comes from the theory of rings of continuous functions. While many real functions are absolutely polynomial, we provide a number of interesting explicit examples which are not. The absolutely polynomial criterion turns out to be quite delicate, and we develop the theory in some detail. Our study of absolutely polynomial functions is then widened to more general topological spaces. Our results provide pertinent counterexamples in the theory of rings of quotients of $\Phi$--algebras.

We characterize the boundedness of fractional maximal functions defined in general homogeneous type spaces from a weighted Banach function spaces $(X,\sg)$ into another $(Y,\om)$ .

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It is shown that every uniform symmetric perfect set admits a minimal Denjoy Index. This minimal Denjoy Index and a point where it is attained is found for many uniform symmetric perfect sets.

We prove that the set $\cal{F}$ of all bounded functionally connected functions is boundary in the space of all bounded Darboux functions (with the metric of uniform convergence). Next we prove that the set of bounded upper (lower)semi-continuous Darboux functions and the set of all bounded quasi-continuous functionally connected functions is porous at each point of the space $\cal{F}$.

For a topological space $X$, let $M(X,R)$ denote the family of all functions $f\in R^{X}$ such that $f(Fr(A))\subseteq Fr(f(A)).$ Let $N(X,R)$ denote the family of all continuous functions $f\in R^{X}$ such that $card(f^{-1}(c))=1$ for each $c\in \Biggl( \inf\limits_{x\in X}f(x),\sup\limits_{x\in X}f(x)\Biggr) .$ We show that $M(X,R)=N(X,R)$ if $X$ is a connected and locally connected Hausdorff space.

In this short note we give a direct proof of a generalization of a standard result due to Havin and Maz\'ya which relates the Bessel capacity of a set to its Hausdorff dimension.

Suppose $(X,d)$ is a compact metric space with the fixed point property and $\,\mathbb{C}\,$ the family of all continuous self maps on $X$ with the topology of uniform convergence. A fixed point $p$ of $ f\in \,\mathbb{C}\,$ is said to be \emph{essential} if functions near $f$ have fixed points near $\,p\,$. A function which has all of its fixed points essential is called an \emph{essential map}. Fort \cite{F} proved that the set of essential maps is residual in $\,\mathbb{C}\,$ and yet the only known examples of essential maps are those with only one fixed point. In this paper working in [0,1], we first characterize essential fixed points and prove some simple results concerning them. Then we characterize essential maps and give algorithms to construct them. We then study essential components introduced by Kinoshita \cite{K}. Next we consider essential fixed points of multifunctions in which case results differ considerably from the case of single valued functions. This also leads us to a study of selections. We conclude with a study of essential fixed points of non expansive functions in Banach spaces. All along we provide examples to illustrate the concepts and their limitations. Our results throw light on what is already known and takes the subject further. Unsolved problems are also mentioned.

Basu and Ganguly recently proved a theorem connected to the classical theorem of Steinhaus which states that $A - B$ has nonempty interior if $A$ and $B$ are Lebesgue measurable subsets of the real line, each having positive measure. The Basu and Ganguly paper deals with a particular 2-place function, namely $f(x,y) = x/y$. There is nothing special about ratios. We will extend their results to functions satisfying simple conditions on their partial derivatives. An $n$ dimensional analogue is also presented.

In this note we prove that, for $a,b \in (0,1)$ and $f$ a measurable function mapping $[0,1]$ to $\R$, the following statements are equivalent: \begin{itemize} \item[(i)] $f(x)=f(x-a)$ a.e.~in $[a,1]$ and $f(x)=f(x-b)$ a.e.~in $[b,1]$ implies that $f$ is a.e.~constant in $[0,1]$. \item[(ii)] $a + b \le 1$ and $a/b$ is irrational. \end{itemize}

We establish conditions for the existence of solutions of the linear integral equation of Volterra \begin{equation} x\left( t\right) +^{\ast }\int\nolimits_{[ a,t] }\alpha ( s) x ( s )\, ds=f ( t ) ,\quad t\in [ a,b ] ,\tag{$V_{\ast}$} \end{equation} where the functions are Banach space-valued and $^{\ast }\int $ denotes either the Bochner-Lebesgue or the Henstock integral. In some cases it is possible to calculate the solution of $( V)_{\ast }$ explicitly. We give several examples.

We prove that the real line is the union of $\co$ disjoint sets each meeting each nonempty set open in the density topology in $\co$ points.

Let the maps $\Lambda $ and $\Omega $ be defined on $C(I,I)$ so that $ f\longmapsto \Lambda (f)=\cup _{x\in I}\omega (x,f)$ and $f\longmapsto \Omega (f)=\{\omega (x,f):x\in I\}.$ We characterize those functions at which $\Lambda $ is continuous, as well as those functions at which $\Omega $ is continuous when its domain is restricted to those elements of $C(I,I)$ possessing zero topological entropy.

In \cite{T13}, Brian S. Thomson proved the following result: \emph{Let $f$ be $AC^*G$ on an interval $[a,b]$. Then the total variation measure $\mu = \mu_f$ associated with $f$ has the following properties: a) $\mu$ is a $\sigma$-finite Borel measure on $[a,b]$; b) $\mu$ is absolutely continuous with respect to Lebesgue measure; \linebreak c) There is a sequence of closed sets $F_n$ whose union is all of $[a,b]$ such that $\mu(F_n) < \infty$ for each $n$; d) $\mu(B) = \mu_f(B) = \int_B|f^\prime(x)|\, dx$ for every Borel set $B \subset [a,b]$. Conversely, if a measure $\mu$ satisfies conditions \linebreak a)--c) then there exists an $AC^*G$ function $f$ for which the representation d) is valid.} In this paper we improve Thomson's theorem as follows: in the first part we ask $f$ to be only $VB^*G \cap (N)$ on a Lebesgue measurable subset $P$ of $[a,b]$ and continuous at each point of $P$; the converse is also true even for $\mu$ defined on the Lebesgue measurable subsets of $P$ (see Theorem \ref{T2} and the two examples in Remark~\ref{R1}).

A type of Strassen's Theorem for measures taking values in the positive cone of a Banach lattice is proved. We generalize a result of A. Hirshberg and R. M. Shortt and formulate a type of Strassen's Theorem in a topological context via closed sets.

In this paper we investigate a certain notion of stability, for one function or for iterated function systems, and discuss why this notion can be a good extension and complement to the notion of hyperbolicity. This last notion is very well-known in the literature and plays an important role in the investigation of the dynamical behavior of a system. The main result is that although some classical sets of functions like the stable Lipschitz functions are conjugate to hyperbolic functions there exist continuous stable functions which are not conjugate to hyperbolic functions. A sufficient condition for not being conjugate to a hyperbolic function is given.

Using some results of M. Laczkovich and T. Keleti we show that for several classes of functions the double difference property holds.

In the present paper we give the Fundamental Theorem of Calculus for the variational or Henstock vector integrals $^{K}\!\!\!\int_{\mathbb R} \alpha \, df$ and $^{K}\!\!\!\int_{\mathbb R} d\alpha \, f$ of multidimensional Banach space-valued functions.

Assuming the continuum hypothesis, there exists a strongly invisible set which is visible from a visible set.

It is shown that if a function $f\colon\real\to\real$ is quasicontinuous and has a graph which is bilaterally dense in itself, then $f$ must be extendable to a connectivity function $F\colon\real^2\to\real$ and the set of discontinuity points of $f$ is $f$-negligible. This improves a result of H.~Rosen. A similar result for symmetrically continuous functions follows immediately.

For each metric space $(X,\xi)$ and each bounded function $f\colon X\to \R$ the family of the sets $\O_f(y)=\{x\in X\colon\, \om_f(x)\ge y\}$ ($\om_f(x)$ is the oscillation of $f$) has some well known properties. In this paper it is constructively shown that for each family $\{\O(y)\}_{y\in [0,1]}$ of subsets of $X$ (separable and $\C$-dense in itself) having similar properties there exists a function $f\colon\, X\to [0,1]$ such that $\O_f(y)=\O(y)$ for each $y\in [0,1]$.

We give corrected proofs of Proposition 2.1 and Theorem 2.3 from \cite{BP}.