Beyond the quasi-Cauchy sequences beyond the Cauchy sequences

In this paper, we investigate the concept of upward continuity. A real valued function on a subset E of R, the set of real numbers is upward continuous if it preserves upward quasi Cauchy sequences in E, where a sequence (x(k)) of points in R is called upward quasi Cauchy if for every epsilon > 0 there exists a positive integer no such that x(n)-x(n+1) < epsilon for n >= n(0). It turns out that the set of upward continuous functions is a proper subset of the set of continuous functions.