Metric spaces on which continuous functions are 'almost' uniformly continuous

摘要

Let X = (X, d), Y = (Y, rho) be two metric spaces and f : X -> Y be a function. f is called almost-bounded iff for every epsilon > 0 there exists a delta > 0 such that for every A subset of X if the diameter of A is < delta, then there is a finite subset B of Y satisfying: For all a is an element of A there exists b is an element of B with f (a) is an element of B(b, epsilon). We show that: