δ-quasi-Cauchy sequences
Mathematical and Computer Modelling, cilt.53, sa.1-2, ss.397-401, 2011 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 53 Sayı: 1-2
- Basım Tarihi: 2011
- Doi Numarası: 10.1016/j.mcm.2010.09.010
- Dergi Adı: Mathematical and Computer Modelling
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
- Sayfa Sayıları: ss.397-401
- Anahtar Kelimeler: Continuity, Real functions, Sequences
- Maltepe Üniversitesi Adresli: Evet
Özet
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. © 2010 Elsevier Ltd.