Suppose f is a function from a metric space (X,D) into another metric space (Y,D') such that D(x,x') >= kD'(f(x),f(x'), where k is a constant positive real number. Prove that f is continuous. Okay, I know that there is a theorem that says "pre-images of open sets are open" so I suppose I can use that. Let U be an open set in (Y,D'). Since U is open, then there exists a neighborhood, N(f(x),p) (a p-neighborhood around f(x).) such that it exists in U. By theorem, neighborhoods are always open. (so basically, I need to find a "q" radius around x such that N(x,q) is open in (X,D).) Although I know what the conclusion should be, I can't find a way to approach that solution. Could anyone give any assistance?