Lacunary Ideal quasi Cauchy sequences

Abstract

A real function is lacunary ideal ward continuous if it preserves lacunary ideal quasi Cauchy sequences where a sequence (x(n)) is said to be lacunary ideal quasi Cauchy (or I-theta-quasi Cauchy) when (Delta x(n)) = (x(n+1) - x(n)) is lacunary ideal convergent to 0. i.e. a sequence (x(n)) of points in R is called lacunary ideal quasi Cauchy (or I-theta-quasi Cauchy) for every epsilon > 0 if

{r is an element of N : 1/hr Sigma(n is an element of Jr) vertical bar x(n+1) - x(n)vertical bar >= epsilon} is an element of I.

Also we introduce the concept of lacunary ideal ward compactness and obtain results related to lacunary ideal ward continuity, lacunary ideal ward compactness, ward continuity, ward compactness, ordinary compactness, uniform continuity, ordinary continuity, delta-ward continuity, and slowly oscillating continuity. Finally we introduce the concept of ideal Cauchy continuous function in metric space and prove some results related to this notion.