δδ-quasi-Cauchy sequences

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