2012 ${N}_{\theta }$-Ward Continuity
Huseyin Cakalli
Abstr. Appl. Anal. 2012: 1-8 (2012). DOI: 10.1155/2012/680456

## Abstract

A function $f$ is continuous if and only if $f$ preserves convergent sequences; that is, $(f({\alpha }_{n}))$ is a convergent sequence whenever $({\alpha }_{n})$ 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, $(f({\alpha }_{n}))$ is an ${N}_{\theta }$-quasi-Cauchy sequence whenever $({\alpha }_{n})$ is ${N}_{\theta }$-quasi-Cauchy. A sequence $({\alpha }_{k})$ of points in $\mathbf{R}$, the set of real numbers, is ${N}_{\theta }$-quasi-Cauchy if ${\mathrm{lim}}_{r\to \infty }(1/{h}_{r}){\sum }_{k\in {I}_{r}}|\Delta {\alpha }_{k}|=0$, where $\Delta {\alpha }_{k}={\alpha }_{k+1}-{\alpha }_{k}$, ${I}_{r}=({k}_{r-1},{k}_{r}],$ and $\theta =({k}_{r})$ 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.

## Citation

Huseyin Cakalli. "${N}_{\theta }$-Ward Continuity." Abstr. Appl. Anal. 2012 1 - 8, 2012. https://doi.org/10.1155/2012/680456

## Information

Published: 2012
First available in Project Euclid: 14 December 2012

zbMATH: 1260.40001
MathSciNet: MR2935154
Digital Object Identifier: 10.1155/2012/680456