  Metric spaces satisfying properties stronger than completeness and weaker
than compactness have been studied by many authors over the years. One such
significant family is that of cofinally complete metric spaces. We discuss the
relationship between cofinally complete metric spaces and the family of almost
uniformly continuous functions, which has recently been introduced by Kyriakos
Keremedis in \cite{[K]}. We also discuss several equivalent conditions for
metric spaces whose completions are cofinally complete in terms of some
geometric functional.
