A study on \(N_\theta\)-quasi-Cauchy sequences. (English) Zbl 1276.26004

Çakalli, Hüseyin; Kaplan, Huseyin

Abstr. Appl. Anal. 2013, Article ID 836970, 4 p. (2013).

Summary: Recently, the concept of \(N_\theta\)-ward continuity was introduced and studied. In this paper, we prove that the uniform limit of \(N_\theta\)-ward continuous functions is \(N_\theta\)-ward continuous, and the set of all \(N_\theta\)-ward continuous functions is a closed subset of the set of all continuous functions. We also obtain that a real function \(f\) defined on an interval \(E\) is uniformly continuous if and only if \((f(\alpha_k))\) is \(N_\theta\)-quasi-Cauchy whenever \((\alpha_k)\) is a quasi-Cauchy sequence of points in \(E\).

MSC:

26A15

Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable

Keywords:

\(N_\theta\)-ward continuous function; quasi-Cauchy sequences

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