Related papers: No function is continuous only at points in a coun…

In 1998, Benyamini introduced and proved the existence of universal interpolating functions. In the note we prove that the set of universal interpolating functions is nowhere dense in the space of continuous functions on $\mathbb{R}$.…

We define and study entanglement of continuous positive definite functions on products of compact groups. We formulate and prove an infinite-dimensional analog of Horodecki Theorem, giving a necessary and sufficient criterion for…

We prove that convex functions of finite order on the real line and subharmonic functions of finite order on finite dimensional real space, bounded from above outside of some set of zero relative Lebesgue density, are bounded from above…

Let $\Gamma$ be a Polish space and let $K$ be a separable and pointwise compact set of real-valued functions on $\Gamma$. It is shown that if each function in $K$ has only countably many discontinuities then $C(K)$ may be equipped with a…

We develop a systematic algorithmic framework that unites global and local classification problems using index sets. We prove that the classification problem for continuous (binary) regular functions among almost everywhere linear,…

In this survey, my aim has been to discuss the use of sequences and countable sets in general topology. In this way I have been led to consider five different classes of topological spaces: first countable spaces, sequential spaces, Frechet…

We give necessary and sufficient conditions for a function in a naturally appearing functional space to be a fixed point of the Ruelle-Thurston operator associated to a rational function, see Lemma 2.1. The proof uses essentially a recent…

We provide several characterizations of the Lebesgue property for fuzzy metric spaces. It is known that a fuzzy metric space is Lebesgue if and only if every real-valued continuous function is uniformly continuous. Here we show that it…

The property of isotonicity of a continuous convex function defined on the entire space or only on the positive cone is characterized via subdifferentials. Numerous examples illustrating the obtained results are included.

We study definably complete locally o-minimal expansions of ordered groups in this paper. A definable continuous function defined on a closed, bounded and definable set behave like a continuous function on a compact set. We demonstrate…

A real function $f$ is ward continuous if $f$ preserves quasi-Cauchyness, i.e. $(f(x_{n}))$ is a quasi-Cauchy sequence whenever $(x_{n})$ is quasi-Cauchy; and a subset $E$ of $\textbf{R}$ is quasi-Cauchy compact if any sequence…

We show that capacity can be computed with locally Lipschitz functions in locally complete and separable metric spaces. Further, we show that if $(X,d,\mu)$ is a locally complete and separable metric measure space, then continuous functions…

Let $\A$ be the operator which assigns to each $m \times n$ matrix-valued function on the unit circle with entries in $H^\infty + C$ its unique superoptimal approximant in the space of bounded analytic $m \times n$ matrix-valued functions…

We show that there is always a uniformly antisymmetric f:A-> {0,1} if A subset R is countable. We prove that the continuum hypothesis is equivalent to the statement that there is an f:R-> omega with |S_x| <= 1 for every x in R. If the…

The main purpose of this paper is to find the fixed point in such cases where existing literature remain silent. In this paper we introduce partial completeness, a new type of contraction and many other definitions. Using this approach the…

We prove a fixed point theorem for closed-graphed, decomposable-valued correspondences whose domain and range is a decomposable set of functions from an atomless measure space to a topological space. One consequence is an improvement of the…

In this note we present a new short and direct proof of L\'{e}vy's continuity theorem in arbitrary dimension $d$, which does not rely on Prohorov's theorem, Helly's selection theorem or the uniqueness theorem for characteristic functions.…

In the paper, we improve our earlier results concerning the existence, uniqueness and differentiability of a global implicit function. Some application to a Cauchy problem for an integro-differential Volterra system of nonconvolution type,…

A topological space $X$ is Baire if the Baire Category Theorem holds for $X$, i.e., the intersection of any sequence of open dense subsets of $X$ is dense in $X$. One of the interesting problems in the theory of functional spaces is the…

In general, some of the well known results of measure theory dealing with the convergence of sequences of functions such as the Dominated Convergence Theorem or the Monotone Convergence Theorem are not true when we consider arbitrary nets…