On Variations of statistical ward continuity

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 ${\lim }_{n\to \infty }\frac{1}{n}|\{k\le n:|{\alpha }_{k+p}-{\alpha }_k|\ge \epsilon \}|=0$ for each $\epsilon >0$. 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 $f(\text{x})=(f({\alpha }_n))$ is statistically $p$-quasi-Cauchy whenever $\mathbf{\alpha }=({\alpha }_n)$ 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 $A$.

13 pages

References in corpus (1)

• Functions preserving slowly oscillating double sequences