Closed & bounded on D imply closed and bounded on f(D)?

The question is:

For D a metric space and f:D--->R a continuous function. If D is closed and bounded. Is f(D) closed and bounded?

I'm pretty sure this is true for the case of D belonging to the real numbers, but I think this is true for D just being a metric space. I think if I can somehow find the example of a metric space that is closed and bounded, but not compact that would be the key. Can't think of one, though. I've also had no luck proving it true. So if someone can give a a counter example or an idea of his to proves, I would greatly appreciate it! Thanks!