Theorem.
Let f be a differentiable function on [a,b] (having at least one-sided derivatives at a and b). Then the derivative f' satisfies the intermediate value theorem on [a,b]. That is, f' attains every value between f'(a) and f'(b) on the interval (a,b).
Note that if f' is continuous, then the theorem follows trivially from the intermediate value theorem. The point of this theorem is that continuity of f' is not necessary!
The theorem is probably more remarkable than its proof.