Intermediate Value Theorem for Derivatives

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
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