Statistical quasi Cauchyness on asymmetric spaces

Abstract

We call a sequence $(x_{m})$ of points in an asymmetric metric space (X,d) statistically forward quasi Cauchy if
$\lim_{n \to \infty }\frac{1}{n}\mid \left \{ m\leq n: \;d(x_{m},x_{m+1})\geq \varepsilon \right \}\mid =0$ for each positive $ \varepsilon$, 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.