On λ statistical upward compactness and continuity

Abstract

A sequence (?k) of real numbers is called ?-statistically upward quasi-Cauchy if for every ? > 0 limn?? 1/?n |{k?In:?k-?k+1 ? ?}| = 0, where (?n) is a non-decreasing sequence of positive numbers tending to 1 such that ?n+1 ? ?n + 1, ?1 = 1, and In = [n-?n+1,n] for any positive integer n. A real valued function f defined on a subset of R, the set of real numbers is ?-statistically upward continuous if it preserves ?-statistical upward quasi-Cauchy sequences. ?-statistically upward compactness of a subset in real numbers is also introduced and some properties of functions preserving such quasi Cauchy sequences are investigated. It turns out that a function is uniformly continuous if it is ?-statistical upward continuous on a ?-statistical upward compact subset of R.