A study on the statistically quasi Cauchy sequences

Abstract

A sequence $(\alpha_{k})$ of real numbers is called ${\lambda}$-statistically upward quasi-Cauchy if for every $\varepsilon>0$
$ \lim_{n\rightarrow\infty}\frac{1}{\lambda_{n} }|\{k\in I_{n}: \alpha_{k}-\alpha_{k+1}\geq \varepsilon\}|=0 $, where $(\lambda_{n})$ is a non-decreasing sequence of positive numbers tending to $\infty$ such that $\lambda_{n+1}\leq \lambda_{n}+1$, $\lambda_{1}=1$, and $I_{n}=[n-\lambda_{n}+1, n]$ for any positive integer $n$. A real valued function $f$ defined on a subset of $\mathbb{R}$, the set of real numbers is $\lambda$-statistically upward continuous if it preserves $\lambda$-statistical upward quasi-Cauchy sequences. It turns out that a function is uniformly continuous if it is $\lambda$-statistical upward continuous on a $\lambda$-statistical upward compact subset of $\mathbb{R}$.