A study on downward half Cauchy sequences

TL;DR

This paper introduces the concepts of down continuity and down compactness, exploring how functions preserve downward half Cauchy sequences, and shows that down continuous functions form a proper subset of continuous functions.

Contribution

It defines and investigates down continuity and down compactness, highlighting their differences from classical continuity and expanding the understanding of sequence preservation.

Findings

Down continuous functions form a proper subset of continuous functions.

Downward half Cauchy sequences are characterized by a specific tail difference condition.

Down continuity is a new form of function continuity based on sequence behavior.

Abstract

In this paper, we introduce and investigate the concepts of down continuity and down compactness. A real valued function $f$ on a subset $E$ of $\R$, the set of real numbers is down continuous if it preserves downward half Cauchy sequences, i.e. the sequence $(f(\alpha_{n}))$ is downward half Cauchy whenever $(\alpha_{n})$ is a downward half Cauchy sequence of points in $E$, where a sequence $(\alpha_{ k})$ of points in $\R$ is called downward half Cauchy if for every $\varepsilon>0$ there exists an $n_{0}\in{\N}$ such that $\alpha_{m}-\alpha_{n} <\varepsilon$ for $m \geq n \geq n_0$. It turns out that the set of down continuous functions is a proper subset of the set of continuous functions.