constant function question

Hi I am having trouble with a proof, was wondering if someone could give me some insight...

Here is the problem:

Suppose f:[a,b] --> Q is continuous on [a,b]. Prove that f is constant on [a,b].

I did a proof by contradiction, but realized that nowhere in my proof was there anything about Q, which Im sure is the crux of the issue.

Any insight into how to prove this using contradiction?

My contradiction was to assume that for some x,y existing in [a,b], f(x) does not equal f(y)....