Ideal quasi-Cauchy sequences

An ideal I is a family of subsets of positive integers N which is closed under taking finite unions and subsets of its elements. A sequence (x(n)) of real numbers is said to be I-convergent to a real number L if for each epsilon > 0, the set {n : vertical bar x(n) - L vertical bar >= epsilon} belongs to I. We introduce I-ward compactness of a subset of R, the set of real numbers, and I-ward continuity of a real function in the senses that a subset E of R is I-ward compact if any sequence (x(n)) of points in E has an I-quasi-Cauchy subsequence, and a real function is I-ward continuous if it preserves I-quasi-Cauchy sequences where a sequence (x(n)) is called to be I-quasi-Cauchy when (Delta x(n)) is I-convergent to 0. We obtain results related to I-ward continuity, I-ward compactness, ward continuity, ward compactness, ordinary compactness, ordinary continuity, delta-ward continuity, and slowly oscillating continuity.