CitationÇakallı, H. (2011). δ-quasi-Cauchy sequences. Mathematical and Computer Modelling. 53(1-2), 397-401.
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 (Δxn) is a null sequence. In this paper we call a real-valued function defined on a subset E of Rδ-ward continuous if it preserves δ-quasi-Cauchy sequences where a sequence x=(xn) is defined to be δ-quasi-Cauchy if the sequence (Δxn) 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.
SourceMathematical and Computer Modelling
- Makale Koleksiyonu