A Variation on Statistical Ward Continuity

A sequence (alpha(k)) of points in R, the set of real numbers, is called rho-statistically convergent to an element l of R if lim (n ->infinity) 1/rho n |{k <= n : |alpha(k)-l| >= epsilon}| = 0 for each epsilon > 0, where rho = (rho n) is a non-decreasing sequence of positive real numbers tending to 8 such that lim sup(n) rho n/n < infinity, Delta rho n = O(1), and Delta alpha(n) = alpha(n+ 1) - alpha(n) for each positive integer n. A real-valued function defined on a subset of R is called rho-statistically ward continuous if it preserves rho-statistical quasi-Cauchy sequences where a sequence (alpha(k)) is defined to be rho-statistically quasi-Cauchy if the sequence (Delta alpha(k)) is rho-statistically convergent to 0. We obtain results related to rho-statistical ward continuity, rho-statistical ward compactness, ward continuity, continuity, and uniform continuity. It turns out that the set of uniformly continuous functions coincides with the set of rho-statistically ward continuous functions not only on a bounded subset of R, but also on an interval.