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The consrtuction of self-similar fuctions in $L_2[0,1]$ is described. Some properties of such funtions (boundness of variation, continuity etc.) is obtained.

We examine conditions on a (compact metrizable) space $X$ such that for any space $Y$ and closed subspace $Z$, the set of continuous functions from $Z$ to $X$ which extend to $Y$ is either open or closed in the set of continuous functions…

If $f : X\mapsto Y$ is a function having Baire property from a metric space $X$ into a separable metric space $Y$ , then $f$ is continuous except on a set of first category. Kuratowski asked whether the condition of separability could be…

Let $X$ denote a Hilbert space. Given a compact subset $K$ of $X$ and two continuous functions $f:K\to\mathbb{R}$, $G:K\to X$, we show that a necessary and sufficient condition for the existence of a convex function $F\in C^1(X)$ such that…

In a paper from 1960, Felix Browder established a theorem concerning the continuation of the fixed points of a family of continuous functions $f_t:X\to X$ depending continuously on a parameter $t\in [0,1]$, where $X$ is a convex and compact…

Using complex methods combined with Baire's Theorem we show that one-sided extendability, extendability and real analyticity are rare phenomena on various spaces of functions in the topological sense. These considerations led us to…

Let $\varphi\colon X\to Y$ be an affine continuous surjection between compact convex sets. Suppose that the canonical copy of the space of real-valued affine continuous functions on $Y$ in the space of real-valued affine continuous…

A fundamental result proved by Bourgain, Fremlin and Talagrand states that the space $B_1(M)$ of Baire one functions over a Polish space $M$ is an angelic space. Stegall extended this result by showing that the class $B_1(M,E)$ of Baire one…

Kechris and Louveau showed that each real-valued bounded Baire class 1 function defined on a compact metric space can be written as an alternating sum of a decreasing countable transfinite sequence of upper semi-continuous functions.…

Let $F:[0,T]\times\R^n\mapsto 2^{\R^n}$ be a continuous multifunction with compact, not necessarily convex values. In this paper, we prove that, if $F$ satisfies the following Lipschitz Selection Property: \begin{itemize} \item[{(LSP)}]…

Let $X$ be a topological space. Let $X_0 \subseteq X$ be a second countable subspace. Also, assume that $X$ is first countable at any point of $X_0$. Then we provide some conditions under which we ensure that $X_0$ is not Baire.

It is a well-known result by Saks \cite{Saks1934} that there exists a function $f \in L^1(\mathbb{R}^2)$ so that for almost every $(x,y)\in \mathbb{R}^2$ \[ \lim_{\substack{\mathrm{diam} R\rightarrow 0, \\ (x,y) \in R \in…

Let $[a,b]\subset\mathbb{R}$ be a non empty and non singleton closed interval and $P=\{a=x_0<\cdots<x_n=b\}$ is a partition of it. Then $f:I\to\mathbb{R}$ is said to be a function of $r$-bounded variation, if the expression…

In this note, we continue to highlight some applications of Theorem 1 of [3]. Here is a sample: Let $X$ be an open set in ${\bf C}^n$, $\Omega$ an open convex set in ${\bf C}$ and $f, g : X\to {\bf C}$ two holomorphic functions such that…

We use lexicographic products to give examples of compact spaces of first Baire class functions on a compact metric space that cannot be represented as spaces of functions with countably many discontinuities.

In this paper we continue to study the property of separability of functional space C(X) with the open-point and bi-point-open topologies.

The \emph{Continuity Problem} is the question whether effective operators are continuous, where an effective operator $F$ is a function on a space of constructively given objects $x$, defined by mapping construction instructions for $x$ to…

We give necessary and sufficient conditions on a function $f:[0,1]\to {0,1,2,...,\omega,\continuum}$ under which there exists a continuous function $F:[0,1]\to [0,1]$ such that for every $y\in[0,1]$ we have $|F^{-1}(y)|=f(y)$.

A separable space is strongly sequentially separable if, for each countable dense set, every point in the space is a limit of a sequence from the dense set. We consider this and related properties, for the spaces of continous and Borel…

Under certain conditions, we obtain sharp bounds on some functionals defined in the coefficient space of starlike functions. It has been found that the functionals are closely associated with certain coefficient problems, which are of…