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 $\mathbb{R}$, 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 $\mathbb{R}$ is called downward half Cauchy if for every $\epsilon >0$ there exists an $n_0\in \mathbb{N}$ such that ${\alpha }_m-{\alpha }_n<\epsilon$ for $m\ge n\ge n_0$. It turns out that the set of down continuous functions is a proper subset of the set of continuous functions.

10 pages

References in corpus (2)

• Functions preserving slowly oscillating double sequences
• On Half Cauchy Sequences