Ideal statistically quasi Cauchy sequences.

An ideal I is a family of subsets of N, the set of positive integers which is closed under taking finite unions and subsets of its elements. A sequence (xk) of real numbers is said to be S(I)-statistically convergent to a real number L, if for each ε > 0 and for each δ > 0 the set {n ∈ N : 1/n |{k ≤ n : |xk -L| ≥ ε }| ≥ δ } belongs to I. We introduce S(I)-statistically ward compactness of a subset of R, the set of real numbers, and S(I)-statistically ward continuity of a real function in the senses that a subset E of R is S(I)-statistically ward compact if any sequence of points in E has an S(I)-statistically quasi- Cauchy subsequence, and a real function is S(I)-statistically ward continuous if it preserves S(I)-statistically quasi-Cauchy sequences where a sequence (xk) is called to be S(I)-statistically quasi-Cauchy when (Δxk) is S(I)-statistically convergent to 0. We obtain results related to S(I)-statistically ward continuity, S(I)-statistically ward compactness, Nθ -ward continuity, and slowly oscillating continuity. [ABSTRACT FROM AUTHOR]