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

On certain notions of precompactness, continuity and Lipschitz functions