Keywords: Sequences, series, summability.

Abstract: A real valued function defined on a subset E of 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→∞}(1/(ρ_{n}))|{k≤n:Δα_{k}≥ε}|=0 for every ε>0, in which (ρ_{n}) is a non-decreasing sequence of positive real numbers tending to ∞ such that limsup_{n}((ρ_{n})/n)<∞, 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.

Download Paper

Share this page