Upward and downward statistical continuities

Abstract

A real valued function f defined on a subset E of R, the set of real numbers, is statistically upward (resp. downward) continuous if it preserves statistically upward (resp. downward) half quasi-Cauchy sequences; A subset E of R, is statistically upward (resp. downward) compact if any sequence of points in E has a statistically upward (resp. downward) half quasi-Cauchy subsequence, where a sequence (xn) of points in R is called statistically upward half quasi-Cauchy if lim n?? 1/n |{k ? n : xk- xk+1 ? ?}| = 0, and statistically downward half quasi-Cauchy if lim n??1/n |{k ? n : xk+1 - xk ? ?}| = 0 for every ? > 0. We investigate statistically upward and downward continuity, statistically upward and downward half compactness and prove interesting theorems. It turns out that any statistically upward continuous function on a below bounded subset of R is uniformly continuous, and any statistically downward continuous function on an above bounded subset of R is uniformly continuous.