Ideal quasi-Cauchy sequences

By Hüseyin Çakalli, Bipan Hazarika. 2012

75 citations recorded by the source corpus.

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 (xn) of real numbers is said to be I-convergent to a real number L if for each ε>0, the set {n:|xn−L|≥ε} 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 (xn) 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 (xn) is called to be I-quasi-Cauchy when (Δxn) is I-convergent to 0. We obtain results related to I-ward continuity, I-ward compactness, ward continuity, ward compactness, ordinary compactness, ordinary continuity, δ-ward continuity, and slowly oscillating continuity.MSC: 40A35, 40A05, 40G15, 26A15.

Subfields: Algebra and Number Theory, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/1c5b7e6ce8a31cc96d8516275d8e29c05e25cf83