On Variations of statistical ward continuity

TL;DR

This paper introduces the concept of statistically p-quasi-Cauchy sequences and statistically p-ward continuous functions, exploring their properties and relationship with uniform continuity on bounded sets.

Contribution

It defines statistically p-quasi-Cauchy sequences and statistically p-ward continuous functions, establishing their connection to uniform continuity on bounded subsets of real numbers.

Findings

Statistically p-quasi-Cauchy sequences are characterized by a specific limit condition.

Statistically p-ward continuous functions preserve these sequences.

Uniform continuity is characterized via the preservation of statistically p-quasi-Cauchy sequences.

Abstract

In this paper, we introduce a concept of statistically $p$-quasi-Cauchyness of a real sequence in the sense that a sequence $(\alpha_{k})$ is statistically $p$-quasi-Cauchy if $\lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: |\alpha_{k+p}-\alpha_{k}|\geq{\varepsilon}\}|=0$ for each $\varepsilon>0$. A function $f$ is called statistically $p$-ward continuous on a subset $A$ of the set of real umbers $\mathbb{R}$ if it preserves statistically $p$-quasi-Cauchy sequences, i.e. the sequence $f(\textbf{x})=(f(\alpha_{n}))$ is statistically $p$-quasi-Cauchy whenever $\boldsymbol\alpha=(\alpha_{n})$ is a statistically $p$-quasi-Cauchy sequence of points in $A$. It turns out that a real valued function $f$ is uniformly continuous on a bounded subset $A$ of $\mathbb{R}$ if there exists a positive integer $p$ such that $f$ preserves statistically $p$-quasi-Cauchy sequences of points in $A$.