delta-quasi-Cauchy sequences
Recently, it has been proved that a real-valued function defined on a subset E of A. 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 (Delta x(n)) is a null sequence. In this paper we call a real-valued function defined on a subset E of R delta-ward continuous if it preserves delta-quasi-Cauchy sequences where a sequence x = (x(n)) is defined to be delta-quasi-Cauchy if the sequence (Delta x(n)) is quasi-Cauchy. It turns out that delta-ward continuity implies uniform continuity, but there are uniformly continuous functions which are not delta-ward continuous. A new type of compactness in terms of delta-quasi-Cauchy sequences, namely delta-ward compactness is also introduced, and some theorems related to delta-ward continuity and delta-ward compactness are obtained. (C) 2010 Elsevier Ltd. All rights reserved.
sequences continuity real functions