Variations on lacunary statistical quasi Cauchy sequences

In this paper, we introduce a concept of lacunary statistically p-quasi-Cauchyness of a real sequence in the sense that a sequence (α_k) is lacunary statistically p-quasi-Cauchy if $limr→∞1hr|{k∈Ir:|αk+p−αk|≥ε}|= 0$ for each ε > 0. A function f is called lacunary statistically p-ward continuous on a subset A of the set of real numbers ℝ if it preserves lacunary statistically p-quasi-Cauchy sequences, i.e. the sequence f (x) = (f (α_n)) is lacunary statistically p-quasi-Cauchy whenever α = (α_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.