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 $\forall \set {a, b \in I: a < b}: \openint a b \subseteq I$, the result follows from Image of Interval by Derivative.
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Source
- 2004: Lars Olsen: A New Proof of Darboux's Theorem (Amer. Math. Monthly Vol. 111, no. 8: pp. 713 – 715) www.jstor.org/stable/4145046