P-Quasi-Cauchy Sequences

Details P-Quasi-Cauchy Sequences P-Quasi-Cauchy Sequences

2012-04-09

arxiv.org/abs/1204.2445v1

Mathematics

General Mathematics

15 pages. arXiv admin note: substantial text overlap with arXiv:1005.4940, arXiv:1203.2003, arXiv:1103.1230, arXiv:1102.1531

Scientific paper

In this paper we generalize the concept of a quasi-Cauchy sequence to a concept of a $p$-quasi-Cauchy sequence for any fixed positive integer $p$. For $p=1$ we obtain some earlier existing results as a special case. We obtain some interesting theorems related to $p$-quasi-Cauchy continuity, $G$-sequential continuity, slowly oscillating continuity, and uniform continuity. It turns out that a function $f$ defined on an interval is uniformly continuous if and only if there exists a positive integer $p$ such that $f$ preserves $p$-quasi-Cauchy sequences where a sequence $(x_{n})$ is called $p$-quasi-Cauchy if $(x_{n+p}-x_{n})_{n=1}^{\infty}$ is a null sequence.

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