A Study on N θ -Quasi-Cauchy Sequences

Abstract

Recently, the concept of $N_{\theta }$-ward continuity was introduced and studied. In this paper, we prove that the uniform limit of $N_{\theta }$-ward continuous functions is $N_{\theta }$-ward continuous, and the set of all $N_{\theta }$-ward continuous functions is a closed subset of the set of all continuous functions. We also obtain that a real function $f$ defined on an interval $E$ is uniformly continuous if and only if ($f$(${\alpha }_k$)) is $N_{\theta }$-quasi-Cauchy whenever (${\alpha }_k$) is a quasi-Cauchy sequence of points in $E$.