Abstract and Applied Analysis

N θ -Ward Continuity

Huseyin Cakalli

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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 θ -ward continuity is defined in the sense that a function f is N θ -ward continuous if it preserves N θ -quasi-Cauchy sequences; that is, ( f ( α n ) ) is an N θ -quasi-Cauchy sequence whenever ( α n ) is N θ -quasi-Cauchy. A sequence ( α k ) of points in R , the set of real numbers, is N θ -quasi-Cauchy if lim r ( 1 / h r ) k I r | Δ α k | = 0 , where Δ α k = α k + 1 - α k , I r = ( k r - 1 , k r ], and θ = ( k r ) is a lacunary sequence, that is, an increasing sequence of positive integers such that k 0 = 0 and h r : k r - k r - 1 . A new type compactness, namely, N θ -ward compactness, is also, defined and some new results related to this kind of compactness are obtained.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 680456, 8 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1355495749

Digital Object Identifier
doi:10.1155/2012/680456

Mathematical Reviews number (MathSciNet)
MR2935154

Zentralblatt MATH identifier
1260.40001

Citation

Cakalli, Huseyin. ${N}_{\theta }$ -Ward Continuity. Abstr. Appl. Anal. 2012 (2012), Article ID 680456, 8 pages. doi:10.1155/2012/680456. https://projecteuclid.org/euclid.aaa/1355495749


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