Upward and downward statistical continuities

Abstract

A real valued  function $f$ defined on a subset $E$ of $\mathbb{R}$, the set of real numbers, is statistically upward continuous if it preserves statistically upward half quasi-Cauchy sequences, is statistically downward continuous if it preserves statistically downward half quasi-Cauchy sequences; and a subset $E$ of $\mathbb{R}$, is statistically upward compact if any sequence of points in $E$ has a statistically upward half quasi-Cauchy subsequence, is statistically downward compact if any sequence of points in $E$ has a statistically downward half quasi-Cauchy subsequence where a sequence $(x_{n})$ of points in $\mathbb{R}$ is called statistically upward half quasi-Cauchy if
\[
\lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: x_{k}-x_{k+1}\geq \varepsilon\}|=0
\]
is statistically downward half quasi-Cauchy if
\[
\lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: x_{k+1}-x_{k}\geq \varepsilon\}|=0
\]
for every $\varepsilon>0$.
We investigate statistically upward continuity, statistically downward continuity, statistically upward half compactness, statistically downward half compactness and prove interesting theorems. It turns out that any statistically upward continuous function on a below bounded subset of $\mathbb{R}$ is uniformly continuous, and any statistically downward continuous function on an above bounded subset of $\mathbb{R}$ is uniformly continuous.