A variation of continuity in $n$-normed spaces

Abstract

In this study, the s-th forward difference sequence that tends to zero, inspired by the consecutive terms of a sequence approaching zero, is examined. Functions that take sequences satisfying this condition to sequences satisfying the same condition are called s-ward continuous. Inclusion theorems related to this kind of continuity and uniform continuity are also considered. In addition, we investigate the concept of $s$-ward compactness of a subset of $X$ via $s$-quasi-Cauchy sequences. It turns out that the uniform limit of a 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.