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

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.