In this paper, we introduce a concept of lacunary statistically p-quasi-Cauchyness of a real sequence in the sense that a sequence (?k) is lacunary statistically p-quasi-Cauchy if limr?? 1 hr |{k ? Ir : |?k+p ? ?k | ? ?}| = 0 for each ? > 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 pquasi-Cauchy sequences, i.e. the sequence f(x) = (f(?n)) is lacunary statistically p-quasi-Cauchy whenever ? = (?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.