Uniformly approachable (UA) functions are a common generalization of uniformly continuous functions an d perfect functions. We study UA-functions and UA-spaces i. e. those uniform spaces in which every real valued continuous function is UA. Such spaces properly include the UC-spaces (Atsuji spaces). We characterize the weakly-UA subspaces of the real line and give a new characterization of the UC spaces. We prove a topological result which implies, under the continuum hypothesis, the existence of a subset M of the the n-dimensional euclidean space R^n such that if two continuous functions f, g from R^n to R are are not constant on any open set and g(M) is a subset of f(M), then f=g.