On sequences in 2-normed spaces

TL;DR

This paper introduces and studies ward continuity in 2-normed spaces, showing how it relates to quasi-Cauchy sequences and demonstrating that uniform limits of ward continuous functions retain this property.

Contribution

It defines ward continuity in 2-normed spaces and explores its properties, including preservation under uniform limits, expanding the understanding of continuity concepts in this setting.

Findings

Ward continuous functions preserve quasi-Cauchy sequences.

Uniform limits of ward continuous functions are also ward continuous.

New types of continuity are introduced via quasi-Cauchy sequences.

Abstract

A function $f$ defined on a 2-normed space $ (X,||.,.||)$ is ward continuous if it preserves quasi-Cauchy sequences where a sequence $(x_n)$ of points in $X$ is called quasi-Cauchy if $lim_{n\rightarrow\infty}||\Delta x_{n},z||=0$ for every $z\in X$. Some other kinds of continuties are also introduced via quasi-Cauchy sequences in 2-normed spaces. It turns out that uniform limit of ward continuous functions is again ward continuous.