turkmath.org

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 $(f (α_n))$ is downward half Cauchy whenever $(α_n)$ is a downward half Cauchy sequence of points in $E$, where a sequence $(α_k )$ of points in $R$ is called downward half Cauchy if for every $ε > 0$ there exists an $n_0 ∈ N$ such that $α_m − α_n < ε$ for $m ≥ n ≥ n_0$. It turns out that the set of down continuous functions is a proper subset of the set of continuous functions.

Keywords: Sequences, series, summability, continuity.
2010 Mathematics Subject Classification: 40A05, 40A35, 40A30, 26A15.