Between continuous and uniformly continuous functions on Rn

We study classes of continuous functions on Rn that can be approximated in various degree by uniformly continuous ones (uniformly approachable functions). It was proved by Berarducci et al. [Topology Appl. 121 (2002)] that no polynomial function can distinguish between them. We construct examples that distinguish these classes (answering a question by Berarducci et al. [Topology Appl. 121 (2002)]) and we offer appropriate forms of uniform approachability that enable us to obtain a general theorem on coincidence in the class of all continuous functions.