Beyond cauchy and quasi-cauchy sequences
Filomat, cilt.32, sa.3, ss.1035-1042, 2018 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 32 Sayı: 3
- Basım Tarihi: 2018
- Doi Numarası: 10.2298/fil1803035c
- Dergi Adı: Filomat
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
- Sayfa Sayıları: ss.1035-1042
- Anahtar Kelimeler: Continuity, Sequences, Series, Summability
- Maltepe Üniversitesi Adresli: Evet
Özet
In this paper, we investigate the concepts of downward continuity and upward continuity. A real valued function on a subset E of ℝ, the set of real numbers, is downward continuous if it preserves downward quasi-Cauchy sequences; and is upward continuous if it preserves upward quasi-Cauchy se- quences, where a sequence (xk) of points in R is called downward quasi-Cauchy if for every ε > 0 there exists an n0 ε ℕ such that xn+1-xn < ε for n ≥ n0, and called upward quasi-Cauchy if for every ε > 0 there exists an n1 ε ℕ such that xn - xn+1 < ε for n ≥ n1. We investigate the notions of downward compactness and upward compactness and prove that downward compactness coincides with above boundedness. It turns out that not only the set of downward continuous functions, but also the set of upward continuous functions is a proper subset of the set of continuous functions.