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

Absolute continuity implies uniform continuity, but generally not vice versa. In this short note, we present one sufficient condition for a uniformly continuous function to be absolutely continuous, which is the following theorem: For a…

We present a constructive proof of Brouwer's fixed point theorem for uniformly continuous and sequentially locally non-constant functions based on the existence of approximate fixed points. And we will show that Brouwer's fixed point…

This paper is a continuation of work started in \cite{njampavcont} on preserving continuity in ideal topological spaces. We will deal with $\theta$-continuity and weak continuity and give their translations in ideal topological spaces. As…

We prove that a set of finite perimeter is indecomposable if and only if it is, up to a choice of suitable representative, connected in the 1-fine topology. This gives a topological characterization of indecomposability which is new even in…

It is known that the topology of a Polish group is uniquely determined by its Borel structure and group operations, but this does not give us a way to find the topology. In this article we expand on this theorem and give a criterion for a…

We prove that every homogeneous countable dense homogeneous topological space containing a copy of the Cantor set is a Baire space. In particular, every countable dense homogeneous topological vector space is a Baire space. It follows that,…

Absolutely continuous commuting row contractions admit a weak-$*$ continuous functional calculus. Building on recent work describing the first and second dual spaces of the closure of the polynomial multipliers on the Drury-Arveson space,…

We prove that there exists a countable family of continuous real functions whose graphs together with their inverses cover an uncountable square, i.e. a set of the form $X\times X$, where $X$ is an uncountable subset of the real line. This…

Let $T$ be a countable complete first-order theory with a definable, infinite, discrete linear order. We prove that $T$ has continuum-many countable models. The proof is purely first-order, but raises the question of Borel completeness of…

In the setting of a metric space equipped with a doubling measure supporting a Poincar\'e inequality, we show that BV functions are, in the sense of multiple limits, continuous with respect to a 1-fine topology, at almost every point with…

For each vector $x\in \ell^{\infty}$, we can define the non-empty compact set $L_x$ of accumulation points of $x$. Given an infinite subset $A$ of $\mathbb{N}\backslash\{1\}$, we can therefore investigate under which conditions on $A$, the…

Let CBV denote the Banach algebra of all continuous real-valued functions of bounded variation, defined in [0,1]. We show that the set of strongly singular functions in CBV is nonseparably spaceable. We also prove that certain families of…

Every computable function has to be continuous. To develop computability theory of discontinuous functions, we study low levels of the arithmetical hierarchy of nonuniformly computable functions on Baire space. First, we classify…

We consider the problem of counting polynomial curves on analytic or definable subsets over the field ${\mathbb{C}}(\!(t)\!)$, as a function of the degree $r$. A result of this type could be expected by analogy with the classical…

If $\mu$ is a finite complex measure in the complex plane $\C$ we denote by $C^\mu$ its Cauchy integral defined in the sense of principal value. The measure $\mu$ is called reflectionless if it is continuous (has no atoms) and $C^\mu=0$ at…

The Fourier series of continuous functions of constant absolute value have interesting properties : according to the main theorems of the article, if the coefficients with positive indexes are square-summable with respect to a certain…

Given a function $f\colon \mathbb{R}\to \mathbb{R}$, the so-called "little lip" function $\operatorname{lip} f$ is defined as follows: \begin{equation*} \operatorname{lip} f(x)=\liminf_{r{\scriptscriptstyle \searrow} 0}\sup_{|x-y|\le r}…

With a new proof approach we prove in a more general setting the classical convergence theorem that almost everywhere convergence of measurable functions on a finite measure space implies convergence in measure. Specifically, we generalize…

In this paper, we introduce the notion of a $\gamma$-density point for Lebesgue-measurable subsets of $\mathbb{R}$, where $\gamma$ is a modulus function, and study its basic measure-theoretic properties. We show that every $\gamma$-density…

A set of points $S$ in Euclidean space $\mathbb{R}^d$ is called \textit{Ramsey} if any finite partition of $\mathbb{R}^{\infty}$ yields a monochromatic copy of $S$. While characterization of Ramsey set remains a major open problem in the…