Abstract
A function is continuous if and only if preserves convergent sequences; that is, is a convergent sequence whenever is convergent. The concept of -ward continuity is defined in the sense that a function is -ward continuous if it preserves -quasi-Cauchy sequences; that is, is an -quasi-Cauchy sequence whenever is -quasi-Cauchy. A sequence of points in , the set of real numbers, is -quasi-Cauchy if , where , and is a lacunary sequence, that is, an increasing sequence of positive integers such that and . A new type compactness, namely, -ward compactness, is also, defined and some new results related to this kind of compactness are obtained.
Highlights
It is well known that a real function f is continuous if and only if, for each point α0 in the domain, limn → ∞f αn f α0 whenever limn → ∞αn α0
A subset A is called G-sequentially closed if it contains all of the points in its G-sequential closure
We recall the concepts of ward compactness, and slowly oscillating compactness: a subset A of R is called ward compact if whenever αn is a sequence of points in A, there is a quasi-Cauchy subsequence z zk αnk of αn 4
Summary
A function f is continuous if and only if f preserves convergent sequences; that is, f αn is a convergent sequence whenever αn is convergent. The concept of Nθ-ward continuity is defined in the sense that a function f is Nθ-ward continuous if it preserves Nθ-quasi-Cauchy sequences; that is, f αn is an Nθ-quasi-Cauchy sequence whenever αn is Nθ-quasi-Cauchy. A new type compactness, namely, Nθ-ward compactness, is defined and some new results related to this kind of compactness are obtained
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