The class of cofinally complete metric spaces lies between the class of complete metric spaces and that of compact metric spaces. It is known that a metric space (X,d) is cofinally complete if and only if every real-valued continuous function on (X,d) is cofinally Cauchy regular, where a function is said to be cofinally Cauchy regular or CC-regular for short if it preserves cofinally Cauchy sequences. Recently in 2017, Keremedis has defined almost bounded functions and AUC spaces [22]. We show that an AUC space is nothing but a cofinally complete metric space and an almost bounded function is nothing but a CC-regular function. Also in this paper, we study boundedness of various Lipschitz-type functions which are CC-regular as well and find equivalent characterizations of metric spaces on which such functions are uniformly continuous. Finally we explore some properties of cofinally Bourbaki Cauchy regular functions, where a function is said to be cofinally Bourbaki Cauchy regular if it preserves cofinally Bourbaki Cauchy sequences [17] and find their relation with CC-regular functions.
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