Intermediate Value Theorem for Derivatives

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Theorem

Let $I$ be an open interval.

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


Then $f'$ satisfies the Intermediate Value Property.


Proof

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

$\blacksquare$