  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
introduce a new 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 along with a very interesting
observation that every real valued ward continuous function defined on a metric
space can be uniformly approximated by real valued quasi-Cauchy Lipschitz
functions.
