In this paper,we introduce concepts of a strongly lacunary p-quasi-Cauchy sequence and strongly lacunary p-ward continuity. We prove that a subset of R is bounded if and only if it is strongly lacunary p-ward compact. It is obtained that any strongly lacunary p-ward continuous function on a subset A of R is continuous in the ordinary sense. We also prove that the uniform limit of strongly lacunary p-ward continuous functions on a subset A of R is strongly lacunary p-ward continuous.