Abel Statistical Quasi Cauchy Sequences

Abstract

In this paper, we investigate the concept of Abel statistical quasi Cauchy sequences. A real function $f$ is called Abel statistically ward continuous it preserves Abel statistical quasi Cauchy sequences, where a sequence $(\alpha_{k})$ of point in $\R$ is called Abel statistically quasi Cauchy if $\lim_{x \to 1^{-}}(1-x)\sum_{k:|\Delta \alpha_{k}|\geq\varepsilon}^{}x^{k}=0$ for every $\varepsilon>0$, where $\Delta \alpha_{k}=\alpha_{k+1}-\alpha_{k}$ for every $k\in{\N}$. Some other types of continuities are also studied and interesting results are obtained. It turns out that the set of Abel statistical ward continuous functions is a closed subset of the space of continuous functions.