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 (formula presented) and statistically downward half quasi-Cauchy if (formula presented) for every ε > 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.