Originally Posted by

**Carl140** Hello guys!

Let (X,d) and (Y,d') be metric spaces and f: X-> Y a continuous map. Suppose that for each a>0 there exists b>0 such that for all x in X we have:

B(f(x), b) is contained in closure( f(B(x,a))).

Here B(f(x),b) represents the open ball with centre f(x) and radius b.

Similarly B(x,a) represents the open ball with centre x and radius a.

Prove that for all x in X and for every c > a :

B(f(x), b) is contained in f(B(x,c)).

I tried to use the characterization of continuity in terms of the closure but got stuck.

Any ideas?