What is the definition of a continuous function on a metric space?
And what are the basic properties?
Definition 1: a function f :X --> Y between metric spaces X,Y with metrix d_X , d_Y is continuous at x in X if for any e > 0 there exists h > 0 s.t. that d_X(x,w) < h ==> d_Y(f(x), f(w)) < e.
Def. 2: a function f: X --> Y between metric spaces X,Y is continuous (in general) if the inverse image of any open ball in X is open in Y
Def. 3: a function F: X --> Y betwen metrix spaces...and etc. is continuous at x in X if for any open ball B in Y containing f(x) there exists an open ball A in X containing x s.t. f(A) is contained in B.
Choose the one you love the most or suits you better.