Cauchy approachable functions and spaces

Abstract

In the context of functions between metric spaces, we introduce a new class of functions which we called Cauchy approachable function which happens to lie between the classes of continuous functions and Cauchy regular functions. A significant characteristic of these functions is that they preserve Cauchy connectedness, despite being a weaker form of Cauchy regularity. We say that a metric space (X, d) is a Cauchy approachable space if every real-valued continuous function on X is Cauchy approachable. Furthermore, by utilizing the concept of Cauchy approachable spaces, we are able to obtain a partial answer to an open question namely, [Question 14.4] posed in [3], in the summary of the last section.