Statistical quasi Cauchyness on asymmetric spaces

We call a sequence (x_m) of points in an asymmetric metric space (X, d) statistically forward quasi Cauchy if limn → ∞ 1/n | \{m ≤ n : d(x_m, 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(x_m)) is statistically forward quasi Cauchy whenever (x_m) is.