Variations on Downard Statistical quasi Cauchy Sequences

Abstract

This work focuses on the introduction and examination of the concepts of statistically p-downward compactness and statistically p-downward continuity. We define a subset E of R, the set of real numbers, to be statistically p-downward compact provided that every sequence in E admits a statistically p-downward quasi-Cauchy subsequence, where a sequence x_k of points in R is called statistically p downward quasi-Cauchy if
\[
\lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: x_{k+p}-x_{k}\geq \varepsilon\}|=0
\]
for every $\varepsilon>0$, where the vertical bars indicate the number of elements in the enclosed set. A real-valued function defined over a subset of R is considered to be statistically p-downward continuous whenever it retains the statistically p-downward quasi-Cauchy nature of sequences.
We investigate statistically p downward continuity, statistically p downward compactness and prove interesting theorems. It turns out that any statistically p downward continuous function on an above bounded subset of R is uniformly continuous and, for any fixed positive integer p, the collection of statistically p-downward continuous functions is strictly contained within the set of all continuous real-valued functions.