We call a sequence (xₘ) of points in an asymmetric metric space (X, d) statistically forward quasi Cauchy if ₍ 1n | \m n: d (xₘ, x₌+₁) \ | = 0 for each positive, 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.
Fikriye Ince Dagci (Wed,) studied this question.