P-Quasi-Cauchy Sequences

TL;DR

This paper introduces the concept of p-quasi-Cauchy sequences, generalizing quasi-Cauchy sequences, and explores their implications for various types of continuity, establishing a characterization of uniform continuity via p-quasi-Cauchy sequence preservation.

Contribution

It extends the theory of quasi-Cauchy sequences to p-quasi-Cauchy sequences and links this to uniform continuity, providing new characterizations and generalizations.

Findings

p-quasi-Cauchy sequences generalize quasi-Cauchy sequences

Uniform continuity characterized by preservation of p-quasi-Cauchy sequences

Established connections with various forms of continuity

Abstract

In this paper we generalize the concept of a quasi-Cauchy sequence to a concept of a $p$-quasi-Cauchy sequence for any fixed positive integer $p$. For $p=1$ we obtain some earlier existing results as a special case. We obtain some interesting theorems related to $p$-quasi-Cauchy continuity, $G$-sequential continuity, slowly oscillating continuity, and uniform continuity. It turns out that a function $f$ defined on an interval is uniformly continuous if and only if there exists a positive integer $p$ such that $f$ preserves $p$-quasi-Cauchy sequences where a sequence $(x_{n})$ is called $p$-quasi-Cauchy if $(x_{n+p}-x_{n})_{n=1}^{\infty}$ is a null sequence.