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 ${\lim }_{n\to \infty }\frac{1}{{\rho }_n}|\{k\le n:\Delta {\alpha }_k\ge \epsilon \}|=0$ for every $\epsilon >0$, in which $({\rho }_n)$ is a non-decreasing sequence of positive real numbers tending to $\infty$ such that ${\mathrm{limsup}}_n\frac{{\rho }_n}{n}<\infty$, $\Delta {\rho }_n=O(1)$, 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.

16 pages. arXiv admin note: text overlap with arXiv:1710.00515