A study on downward half Cauchy sequences

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.