I'm really stuck on this problem...

If I let the function:

f : I→ I be continuous on I and differentiable on the open set I

for I := [0,1]

I know that theresat least1 point t ∈ [0, 1] such that f(t) = t.

Now I need to use Rolle’s Theorem to show that if f'(x) is not equal to 1 in (0, 1), then there isexactly onesuch point t

I've tried 3 different proofs for this, but none of them are giving me uniqueness of t. Please help!