Abel Continuity

TL;DR

This paper introduces Abel continuity, a new form of function continuity based on Abel convergence of sequences, and explores its properties, related compactness concepts, and connections to other types of continuity.

Contribution

It defines Abel continuity and Abel sequential compactness, establishing their fundamental properties and relationships with existing continuity notions.

Findings

Abel continuous functions preserve Abel convergent sequences.

Introduces Abel sequential compactness as a new compactness concept.

Establishes links between Abel continuity and other continuity types.

Abstract

A sequence $\textbf{p}=(p_{n})$ of real numbers is called Abel convergent to $\ell$ if the series $\Sigma_{k=0}^{\infty}p_{k}x^{k}$ is convergent for $0\leq x<1$ and \[\lim_{x \to 1^{-}}(1-x) \sum_{k=0}^{\infty}p_{k}x^{k}=\ell.\] We introduce a concept of Abel continuity in the sense that a function $f$ defined on a subset of $\Re$, the set of real numbers, is Abel continuous if it preserves Abel convergent sequences, i.e. $(f(p_{n}))$ is an Abel convergent sequence whenever $(p_{n})$ is. A new type of compactness, namely Abel sequential compactness is also introduced, and interesting theorems related to this kind of compactnes, Abel continuity, statistical continuity, lacunary statistical continuity, ordinary continuity, and uniform continuity are obtained.