Half quasi-Cauchy sequences

TL;DR

This paper introduces and studies the concepts of upward and downward half quasi-Cauchy continuity and compactness, extending existing ideas of ward continuity and quasi-Cauchy compactness in real analysis.

Contribution

It defines new types of half quasi-Cauchy continuity and compactness, and investigates their properties and relationships, expanding the framework of quasi-Cauchy sequence analysis.

Findings

Established properties of upward and downward half quasi-Cauchy continuous functions.

Proved theorems relating half quasi-Cauchy compactness to other forms of compactness.

Extended the theory of quasi-Cauchy sequences with new directional concepts.

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 $\textbf{R}$ is quasi-Cauchy compact if any sequence $\textbf{x}=(x_{n})$ of points in $E$ has a quasi-Cauchy subsequence where $\textbf{R}$ 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 $\textbf{R}$ 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.