**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.