A new variation on statistical ward continuity

Abstract

A real valued function defined on a subset $E$ of $\mathbb{R}$, the set of real numbers, is $\rho$-statistically downward continuous if it preserves $\rho$-statistical downward quasi-Cauchy sequences of points in $E$, where a sequence $(\alpha_{k})$ of real numbers is called ${\rho}$-statistically downward quasi-Cauchy if $\lim_{n\rightarrow\infty}\frac{1}{\rho_{n} }|\{k\leq n: \Delta \alpha_{k} \geq \varepsilon\}|=0 $ for every $\varepsilon>0$, in which $(\rho_{n})$ is a non-decreasing sequence of positive real numbers tending to $\infty$ such that $\limsup _{n} \frac{\rho_{n}}{n}<\infty $, $\Delta \rho_{n}=O(1)$, and $\Delta \alpha _{k} =\alpha _{k+1} - \alpha _{k}$ for each positive integer $k$. It turns out that a function is uniformly continuous if it is $\rho$-statistical downward continuous on an above bounded set.