Characterization for Cauchy Continuity and Sufficiency of Sequences

https://doi.org/10.65770/FUKV4767

ABSTRACT

A function from a metric space to a metric space preserves Cauchy sequences if and only if it is uniformly continuous on every totally bounded subset of the domain; or equivalently, if and only if it is uniformly continuous on every subset whose members are the terms of a Cauchy sequence in the domain.  This characterization is established in this article.  Possible compromises to replace nets by sequences are also discussed.  A linear mapping from a metrizable topological vector space to a metrizable topological vector space is continuous if and only if it preserves Cauchy sequences.  This fact is also observed as an application of the first characterization.

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WSN 218 (2026) 44-55

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