Slowly oscillating continuity
Abstract and Applied Analysis, cilt.2008, 2008 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 2008
- Basım Tarihi: 2008
- Doi Numarası: 10.1155/2008/485706
- Dergi Adı: Abstract and Applied Analysis
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
- Maltepe Üniversitesi Adresli: Evet
Özet
A function f is continuous if and only if, for each point x 0 in the domain, lim n→∞ f(x n)=f(x 0), whenever lim n→∞ x n = x 0. This is equivalent to the statement that (f(x n)) is a convergent sequence whenever (x n) is convergent. The concept of slowly oscillating continuity is defined in the sense that a function f is slowly oscillating continuous if it transforms slowly oscillating sequences to slowly oscillating sequences, that is, (f(x n)) is slowly oscillating whenever (x n) is slowly oscillating. A sequence (x n) of points in R is slowly oscillating if lim λ → 1 + lim n max n + 1 ≤ k ≤ [ λ n ] | x k - x n | = 0, where [ λn ] denotes the integer part of λn. Using ε>0 's and δ 's, this is equivalent to the case when, for any given ε>0, there exist δ=δ(ε)>0 and N=N(ε) such that | x m - x n |<ε if n≥N(ε) and n≤m≤(1+δ)n. A new type compactness is also defined and some new results related to compactness are obtained.