Variations on lacunary statistical quasi Cauchy sequences

Abstract

In this paper, we introduce a concept of lacunary statistically p-quasi-Cauchyness of a real sequence in the sense that a sequence (alpha(k)) is lacunary statistically p-quasi-Cauchy if lim(r ->infinity) 1/h(r)vertical bar{k is an element of I-r : vertical bar alpha(k+p) - alpha(k)vertical bar >= epsilon}vertical bar = 0 for each epsilon > 0. A function f is called lacunary statistically p-ward continuous on a subset A of Me set of real numbers R if it preserves lacunary statistically p quasi-Cauchy sequences, i.e. the sequence f(x) = (f(alpha(n))) is lacunary statistically p-quasi-Cauchy whenever alpha = (alpha(n)) is a lacunary statistically p-quasi-Cauchy sequence of points in A. It turns out that a real valued function f is uniformly continuous on a bounded subset A of R if there exists a positive integer p such that f preserves lacunary statistically p-quasi-Cauchy sequences of points in A.