Prove that the Image of a Continuous Function on a Dense Set is Dense

I've been working on this for days and have made no progress, so any advice would be appreciated.

Definition: Let be an ordered field, and . is dense in iff for any in there is a such that .

Prove that, for any function continuous on a metric space and a dense subset, then is dense.

So far what I have is hardly more than setting the problem up. Let . Thus (for now, lets assume ), and by density of , such that . I know that I will be basically done if I can show that and . I also know that the inverse map must also be continuous. But that's all I got.

Thanks in advance.