Functions that preserve totally bounded sets vis-\'a-vis stronger notions of continuity

TL;DR

This paper investigates functions that preserve totally bounded sets in metric spaces, comparing their properties with various forms of continuity and analyzing their stability under reciprocation.

Contribution

It introduces and studies totally bounded regular functions, examining their relation to continuous and stronger classes of functions, including strongly uniformly continuous functions.

Findings

Totally bounded regular functions are not necessarily continuous.

Strongly uniformly continuous functions are analyzed for stability under reciprocation.

The paper characterizes conditions under which these functions preserve certain boundedness properties.

Abstract

A function between two metric spaces is said to be totally bounded regular if it preserves totally bounded sets. These functions need not be continuous in general. Hence the purpose of this article is to study such functions vis-\'a-vis continuous functions and functions that are stronger than the continuous functions such as Cauchy continuous functions, some Lipschitz-type functions etc. We also present some analysis on strongly uniformly continuous functions which were first introduced in \cite{[BL2]} and study when these functions are stable under reciprocation.