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Abstract and Applied Analysis
Volume 2008, Article ID 485706, 5 pages
http://dx.doi.org/10.1155/2008/485706
Research Article

Slowly Oscillating Continuity

Department of Mathematics, Faculty of Science and Letters, Maltepe University, 34857 Maltepe, Istanbul, Turkey

Received 2 November 2007; Accepted 11 February 2008

Academic Editor: Ferhan Atici

Copyright © 2008 H. Çakalli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A function 𝑓 is continuous if and only if, for each point π‘₯ 0 in the domain, l i m 𝑛 β†’ ∞ 𝑓 ( π‘₯ 𝑛 ) = 𝑓 ( π‘₯ 0 ) , whenever l i m 𝑛 β†’ ∞ π‘₯ 𝑛 = π‘₯ 0 . This is equivalent to the statement that ( 𝑓 ( π‘₯ 𝑛 ) ) is a convergent sequence whenever ( π‘₯ 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 ( π‘₯ 𝑛 ) ) is slowly oscillating whenever ( π‘₯ 𝑛 ) is slowly oscillating. A sequence ( π‘₯ 𝑛 ) of points in 𝐑 is slowly oscillating if l i m πœ† β†’ 1 + l i m 𝑛 m a x 𝑛 + 1 ≀ π‘˜ ≀ [ πœ† 𝑛 ] | π‘₯ π‘˜ βˆ’ π‘₯ 𝑛 | = 0 , where [ πœ† 𝑛 ] denotes the integer part of πœ† 𝑛 . Using πœ€ > 0 's and 𝛿 's, this is equivalent to the case when, for any given πœ€ > 0 , there exist 𝛿 = 𝛿 ( πœ€ ) > 0 and 𝑁 = 𝑁 ( πœ€ ) such that | π‘₯ π‘š βˆ’ π‘₯ 𝑛 | < πœ€ if 𝑛 β‰₯ 𝑁 ( πœ€ ) and 𝑛 ≀ π‘š ≀ ( 1 + 𝛿 ) 𝑛 . A new type compactness is also defined and some new results related to compactness are obtained.