Cauchy-subregular functions vis-à-vis different types of continuity

Abstract

A function between two metric spaces is said to be Cauchy-regular if it takes Cauchy sequences to Cauchy sequences. This well-studied class of functions lies strictly in between the class of continuous functions and that of uniformly continuous functions. In this paper, we exhibit another class of functions between metric spaces which is a weaker form of Cauchy-regular functions, called Cauchy-subregular functions. We call a function f:(X,d)→(Y,ρ) to be Cauchy-subregular if for every Cauchy sequence (xn) in (X,d), (f(xn)) has a Cauchy subsequence. We investigate how much it differs from Cauchy-regularity, while using this weaker notion profitably to characterize complete metric spaces. We also compare Cauchy-subregular functions to various other types of continuity such as locally Lipschitz functions and subcontinuous functions. Furthermore, we study relation between Cauchy-subregular functions and the functions that preserve cofinally Cauchy sequences called cofinally Cauchy regular functions.