A variation of continuity in $n$-normed spaces

TL;DR

This paper introduces and studies the concept of s-ward continuity and s-ward compactness in n-normed spaces, exploring their properties, inclusion relations, and the behavior of limits of such functions.

Contribution

It defines s-ward continuity and compactness in n-normed spaces, establishing their fundamental properties and relationships with classical continuity concepts.

Findings

Uniform limits of s-ward continuous functions are s-ward continuous.

The set of s-ward continuous functions forms a closed subset of continuous functions.

Inclusion theorems relate s-ward continuity to other forms of continuity.

Abstract

The s-th forward difference sequence that tends to zero, inspired by the consecutive terms of a sequence approaching zero, is examined in this study. Functions that take sequences satisfying this condition to sequences satisfying the same condition are called s-ward continuous. Inclusion theorems that are related to this kind of uniform continuity and continuity are also considered. Additionally, the concept of $s$-ward compactness of a subset of $X$ via $s$-quasi-Cauchy sequences are investigated. One finds out that the uniform limit of any sequence of $s$-ward continuous function is $s$-ward continuous and the set of $s$-ward continuous functions is a closed subset of the set of continuous functions.