Abstract and Applied Analysis

Slowly Oscillating Continuity

H. Çakalli

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Abstract

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 [ λ n ] denotes the integer part of λ n . Using ɛ > 0 's and δ 's, this is equivalent to the case when, for any given ɛ > 0 , there exist δ = δ ( ɛ ) > 0 and N = N ( ɛ ) such that | x m x n | < ɛ if n N ( ɛ ) and n m ( 1 + δ ) n . A new type compactness is also defined and some new results related to compactness are obtained.

Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 485706, 5 pages.

Dates
First available in Project Euclid: 9 September 2008

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1220969159

Digital Object Identifier
doi:10.1155/2008/485706

Mathematical Reviews number (MathSciNet)
MR2393124

Zentralblatt MATH identifier
1153.26002

Citation

Çakalli, H. Slowly Oscillating Continuity. Abstr. Appl. Anal. 2008 (2008), Article ID 485706, 5 pages. doi:10.1155/2008/485706. https://projecteuclid.org/euclid.aaa/1220969159


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