1. ## Continuity characterization help

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?

2. 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?
Since $f$ is continuous, for all $E \subseteq X$, $f(\overline{E}) \subseteq \overline{f(E)}$. Now we know that $B(f(x),b) \subseteq \overline{f(B(x,a))}$. We want to show that $B(f(x),b) \subseteq f(B(x,c))$ for every $c>a$. I think it boils down to showing that $\overline{f(B(x,a))} \subseteq f(B(x,c))$.

3. Right, I have tried to do that, but got stuck. Can you please help a little bit?