On certain notions of precompactness, continuity and Lipschitz functions

The underlying theme of this article is a class of sequences in metric structures satisfying a much weaker kind of Cauchy condition, namely quasi-Cauchy sequences (introduced in \cite{bc}) that has been used to define several new concepts in recent articles \cite{PDSPNA2, PDSPNA1}. We first consider a weaker notion of precompactness based on the idea of quasi-Cauchy sequences and establish several results including a new characterization of compactness in metric spaces. Next we consider associated idea of continuity, namely, ward continuous functions \cite{caka}, as this class of functions strictly lies between the classes of continuous and uniformly continuous functions and mainly establish certain coincidence results. Finally a new class of Lipschitz functions called "quasi-Cauchy Lipschitz functions" is introduced following the line of investigations in \cite{Beer1,Beer2,Beer3,g1} and again several coincidence results are proved. The motivation behind such kind of Lipschitz functions is ascertained by the observation that every real valued ward continuous function defined on a metric space can be uniformly approximated by real valued quasi-Cauchy Lipschitz functions.