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2008 Slowly Oscillating Continuity
H. Çakalli
Abstr. Appl. Anal. 2008: 1-5 (2008). DOI: 10.1155/2008/485706

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.

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H. Çakalli. "Slowly Oscillating Continuity." Abstr. Appl. Anal. 2008 1 - 5, 2008. https://doi.org/10.1155/2008/485706

Information

Published: 2008
First available in Project Euclid: 9 September 2008

zbMATH: 1153.26002
MathSciNet: MR2393124
Digital Object Identifier: 10.1155/2008/485706

Rights: Copyright © 2008 Hindawi

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