N θ -Ward Continuity

Abstract

A function $f$ is continuous if and only if $f$ preserves convergent sequences; that is, $\left(f\right({\alpha }_n\left)\right)$ is a convergent sequence whenever $\left({\alpha }_n\right)$ is convergent. The concept of $N_{\theta }$-ward continuity is defined in the sense that a function $f$ is $N_{\theta }$-ward continuous if it preserves $N_{\theta }$-quasi-Cauchy sequences; that is, $\left(f\right({\alpha }_n\left)\right)$ is an $N_{\theta }$-quasi-Cauchy sequence whenever $\left({\alpha }_n\right)$ is $N_{\theta }$-quasi-Cauchy. A sequence $\left({\alpha }_k\right)$ of points in $\mathit{R}$, the set of real numbers, is $N_{\theta }$-quasi-Cauchy if ${\lim }_{r\to \infty }(1/h_r){\sum }_{k\in I_r}\left|\Delta {\alpha }_k\right|=0$, where $\Delta {\alpha }_k={\alpha }_{k+1}-{\alpha }_k$, $I_r=(k_{r-1},k_r],$ and $\theta =\left(k_r\right)$ is a lacunary sequence, that is, an increasing sequence of positive integers such that $k_0=0$ and $h_r:k_r-k_{r-1}\to \infty$. A new type compactness, namely, $N_{\theta }$-ward compactness, is also, defined and some new results related to this kind of compactness are obtained.