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