Intermediate Value Theorem for Derivatives

Theorem

Let $I$ be an open interval.

Let $f : I \rightarrow \R$ be everywhere differentiable.

Then $f'$ satisfies the Intermediate Value Property.

Proof

Since $\forall \left[a,b \in I: a < b\right]: (a..b) \subseteq I$, the result follows from Image of Interval by Derivative.

$\blacksquare$