# Prove that f must be a constant function

1. Oct 3, 2011

### SithsNGiggles

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. 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.

2. Oct 3, 2011

### 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?

3. Oct 3, 2011

### daveb

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

4. Oct 3, 2011

### gb7nash

Are you allowed to use intermediate value theorem?

5. Oct 3, 2011

### SithsNGiggles

Oh, ok I think I got it. Thanks.