Formula omitted]-quasi-Cauchy sequences

Abstract

Recently, it has been proved that a real-valued function defined on a subset E of R , the set of real numbers, is uniformly continuous on E if and only if it is defined on E and preserves quasi-Cauchy sequences of points in E where a sequence is called quasi-Cauchy if ( Δ x n ) is a null sequence. In this paper we call a real-valued function defined on a subset E of R δ -ward continuous if it preserves δ -quasi-Cauchy sequences where a sequence x = ( x n ) is defined to be δ -quasi-Cauchy if the sequence ( Δ x n ) is quasi-Cauchy. It turns out that δ -ward continuity implies uniform continuity, but there are uniformly continuous functions which are not δ -ward continuous. A new type of compactness in terms of δ -quasi-Cauchy sequences, namely δ -ward compactness is also introduced, and some theorems related to δ -ward continuity and δ -ward compactness are obtained.