On Variations of statistical ward continuity

In this paper, we introduce a concept of statistically $p$-quasi-Cauchyness of a real sequence in the sense that a sequence $({\alpha }_k)$ is statistically $p$-quasi-Cauchy if $\lim_{n\rightarrow\infty}\frac{1}{n}|{k\leq n: |\alpha_{k+p}-\alpha_{k}|\geq{\varepsilon}}|=0$foreach$\varepsilon>0$. A function $f$ is called statistically $p$-ward continuous on a subset $A$ of the set of real umbers $\mathbb{R}$ if it preserves statistically $p$-quasi-Cauchy sequences, i.e. the sequence $f(\mathbf{\text{x}})=(f({\alpha }_n))$ is statistically $p$-quasi-Cauchy whenever $\mathit{\alpha }=({\alpha }_n)$ is a statistically $p$-quasi-Cauchy sequence of points in $A$. It turns out that a real valued function $f$ is uniformly continuous on a bounded subset $A$ of $\mathbb{R}$ if there exists a positive integer $p$ such that $f$ preserves statistically $p$-quasi-Cauchy sequences of points in $A$.