A study on downward half Cauchy sequences
Huseyin Cakalli

TL;DR
This paper introduces the concepts of down continuity and down compactness, exploring how functions preserve downward half Cauchy sequences, and shows that down continuous functions form a proper subset of continuous functions.
Contribution
It defines and investigates down continuity and down compactness, highlighting their differences from classical continuity and expanding the understanding of sequence preservation.
Findings
Down continuous functions form a proper subset of continuous functions.
Downward half Cauchy sequences are characterized by a specific tail difference condition.
Down continuity is a new form of function continuity based on sequence behavior.
Abstract
In this paper, we introduce and investigate the concepts of down continuity and down compactness. A real valued function on a subset of , the set of real numbers is down continuous if it preserves downward half Cauchy sequences, i.e. the sequence is downward half Cauchy whenever is a downward half Cauchy sequence of points in , where a sequence of points in is called downward half Cauchy if for every there exists an such that for . It turns out that the set of down continuous functions is a proper subset of the set of continuous functions.
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Taxonomy
TopicsApproximation Theory and Sequence Spaces · Mathematical Analysis and Transform Methods · Advanced Banach Space Theory
