Darboux's Theorem (Intermediate Value Property of Derivative on Closed Interval)

Let I be a closed interval and f : I → ℝ be a differentiable function. Then the derivative f' of f has the intermediate value property: if a,b ∈ I with a < b, then for every y ∈ ℝ such that 

f'(a) < y < f'(b) or f'(b) < y < f'(a), there exists x ∈ [a,b] such that f'(x) = y.