Munkres "Analysis on Manifolds "P. 30 #5 asks: let f: X- > Y. Show that f is continuous if and only if for each x [in] X there is a neighborhood U of x such that flU is continuous. I can understand I think how to prove the condition is necessary. But since x is in U doesn't the condition state "f is continuous for all x in X?" (so why need to prove the condition is sufficient?)