Question

darboux theorem

Answer 1

Answer on Question # 41300 – Math - Real Analysis

Darboux theorem

Answer:

Let $I$ be an open interval, $f\colon I \to \mathbb{R}$ a real-valued differentiable function. Then $f'$ has the intermediate value property: If $a$ and $b$ are points in $I$ with $a \leq b$, then for every $k$ between $f'(a)$ and $f'(b)$, there exists an $c$ in $[a, b]$ such that $f'(c) = k$.

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