Beyond Cauchy and Quasi-Cauchy Sequences
In this paper, we investigate the concepts of downward continuity and upward continuity. A real valued function on a subset E of 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 R is called downward quasi-Cauchy if for every epsilon > 0 there exists an n(0) is an element of N such that x(n+1) xn < epsilon for n >= n(0), and called upward quasi-Cauchy if for every epsilon > 0 there exists an n(1) is an element of N such that xn x(n+1) < epsilon for n >= 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.