Prove that f must be a constant function

[SithsNGiggles]

Homework Statement

If f: ℝ → ℝ is a continuous function with the property that its range is contained in the set of integers, prove that f must be a constant function.

Homework Equations

The Attempt at a Solution

I know why this is true, I just don't know how to begin an actual proof. So far I've thought of proving by contradiction, with letting f be discontinuous and use f(x) = [x], whose range set is contained in Z.

I seem to have trouble with the format of a formal proof.

 

[Robert1986]

OK, suppose f(x) and f(y) are not equal for some x and y. Now, you know that their difference must be at least 1, right? So, let $\epsilon = 1$ and...

Do you see how this might work?

 

[daveb]

It seems an epsilon-delta proof might work well here.

 

[gb7nash]

Are you allowed to use intermediate value theorem?

 

[SithsNGiggles]

Oh, ok I think I got it. Thanks.