Quasi-Cauchy Sequences, the Functions that Preserve them, and a Weakening of Bourbaki Boundedness

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 that has been used to define several new concepts in recent articles due to Das, Pal, and Adhikary. 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, 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 Beer and Garrido (Topology Appl., 208 (2016), 1-9) 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.