Statistical Forward Continuity in Asymmetric Metric Spaces

Abstract

A sequence (xm) of points in an asymmetric metric space X is called statistically forward convergent to a point L of X if limn→∞1n|{m≤n:d(L,xm)≥ϵ}|=0 and is called quasi Cauchy if limn→∞1n|{m≤n:d(L,xm+1)≥ϵ}|=0 for each positive ϵ, where |A| indicates the cardinality of the set A. We prove that a subset E of X is forward totally bounded if and only if any sequence of points in E has a statistically forward quasi Cauchy subsequence. We also introduce and investigate statistically upward continuity in the sense that a function defined on X into Y is called statistically upward continuous if it preserves statistically forward quasi Cauchy sequences, i.e. (f (xm)) is statistically forward quasi Cauchy whenever (xm) is. © 2025 Author(s).