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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 680456, 8 pages
http://dx.doi.org/10.1155/2012/680456
Research Article

𝑁 πœƒ -Ward Continuity

Department of Mathematics, Maltepe University, Marmara Education Village, 34857 Istanbul, Turkey

Received 8 March 2012; Accepted 19 April 2012

Academic Editor: Ljubisa Kocinac

Copyright © 2012 Huseyin Cakalli. 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 𝑓 preserves convergent sequences; that is, ( 𝑓 ( 𝛼 𝑛 ) ) is a convergent sequence whenever ( 𝛼 𝑛 ) is convergent. The concept of 𝑁 πœƒ -ward continuity is defined in the sense that a function 𝑓 is 𝑁 πœƒ -ward continuous if it preserves 𝑁 πœƒ -quasi-Cauchy sequences; that is, ( 𝑓 ( 𝛼 𝑛 ) ) is an 𝑁 πœƒ -quasi-Cauchy sequence whenever ( 𝛼 𝑛 ) is 𝑁 πœƒ -quasi-Cauchy. A sequence ( 𝛼 π‘˜ ) of points in 𝐑 , the set of real numbers, is 𝑁 πœƒ -quasi-Cauchy if l i m π‘Ÿ β†’ ∞ ( 1 / β„Ž π‘Ÿ ) βˆ‘ π‘˜ ∈ 𝐼 π‘Ÿ | Ξ” 𝛼 π‘˜ | = 0 , where Ξ” 𝛼 π‘˜ = 𝛼 π‘˜ + 1 βˆ’ 𝛼 π‘˜ , 𝐼 π‘Ÿ = ( π‘˜ π‘Ÿ βˆ’ 1 , π‘˜ π‘Ÿ ] , and πœƒ = ( π‘˜ π‘Ÿ ) is a lacunary sequence, that is, an increasing sequence of positive integers such that π‘˜ 0 = 0 and β„Ž π‘Ÿ ∢ π‘˜ π‘Ÿ βˆ’ π‘˜ π‘Ÿ βˆ’ 1 β†’ ∞ . A new type compactness, namely, 𝑁 πœƒ -ward compactness, is also, defined and some new results related to this kind of compactness are obtained.