On Abel statistical convergence

TL;DR

This paper introduces the concept of Abel statistical continuity, studying functions that preserve Abel statistically convergent sequences and exploring related types of continuity with new results.

Contribution

It defines Abel statistical continuity and investigates its properties, establishing its relationship with other forms of continuity in real analysis.

Findings

Abel statistical continuous functions preserve Abel statistical convergence.

The paper establishes properties and relationships with other continuity types.

New results on the behavior of Abel statistical continuous functions are presented.

Abstract

In this paper, we introduce and investigate a concept of Abel statistical continuity. A real valued function $f$ is Abel statistically continuous on a subset $E$ of $\R$, the set of real numbers, if it preserves Abel statistical convergent sequences, i.e. $(f(p_{k}))$ is Abel statistically convergent whenever $(p_{k})$ is an Abel statistical convergent sequence of points in $E$, where a sequence $(p_{k})$ of point in $\R$ is called Abel statistically convergent to a real number $L$ if Abel density of the set $\{k\in{\N}: |p_{k}-L|\geq\varepsilon \}$ is $0$ for every $\varepsilon>0$. Some other types of continuities are also studied and interesting results are obtained.