Half quasi-Cauchy sequences

Abstract

A real function $f$ is ward continuous if $f$ preserves quasi-Cauchyness, i.e. $(f(x_n))$ is a quasi-Cauchy sequence whenever $(x_n)$ is quasi-Cauchy; and a subset $E$ of $\text{R}$ is quasi-Cauchy compact if any sequence $\text{x}=(x_n)$ of points in $E$ has a quasi-Cauchy subsequence where $\text{R}$ is the set of real numbers. These known results suggest to us introducing a concept of upward (respectively, downward) half quasi-Cauchy continuity in the sense that a function $f$ is upward (respectively, downward) half quasi-Cauchy continuous if it preserves upward (respectively, downward) half quasi-Cauchy sequences, and a concept of upward (respectively, downward) half quasi-Cauchy compactness in the sense that a subset $E$ of $\text{R}$ is upward (respectively, downward) half quasi-Cauchy compact if any sequence of points in $E$ has an upward (respectively, downward) half quasi-Cauchy subsequence. We investigate upward(respectively, downward) half quasi-Cauchy continuity and upward (respectively, downward) half quasi-Cauchy compactness, and prove related theorems.

24 pages. arXiv admin note: substantial text overlap with arXiv:1103.1230, arXiv:1102.1531, arXiv:1204.2445

References in corpus (1)

• On Half Cauchy Sequences