Beyond Cauchy and Quasi-Cauchy Sequences

Abstract

In this paper, we investigate the concepts of downward continuity and upward continuity. A real valued function on a subset $E$ of $\mathbb{R}$, the set of real numbers is downward continuous if it preserves downward quasi Cauchy sequences; and is upward continuous if it preserves upward  quasi Cauchy sequences, where a sequence $(x_{ k})$ of points in $\mathbb{R}$ is called downward   quasi Cauchy if for every $\varepsilon>0$ there exists an $n_{0}\in{\mathbb{N}}$ such that $x_{n+1}-x_{n} <\varepsilon$ for $n \geq n_{0}$, and called upward  quasi Cauchy if for every $\varepsilon>0$ there exists an $n_{1}\in{\mathbb{N}}$ such that $x_{n}-x_{n+1} <\varepsilon$ for $n \geq n_{1}$. We investigate the notions of downward compactness and upward compactness and prove that downward compactness coincides with above boundedness.  It turns out that not only the set of downward continuous  functions, but also the set of upward continuous functions is a proper subset of the set of continuous functions