Functions that preserve certain classes of sequences and locally Lipschitz functions

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.