Darboux's theorem

Darboux’s theorem: Let $I\subseteq\mathbb{R}$ be an interval , and let $f:I\to\mathbb{R}$ be differentiable on $I$. Then the derivative $f'$ has the intermediate value property: whenever $a<b$ are in $I$ and $y$ lies between $f'(a)$ and $f'(b)$, there exists $c\in(a,b)$ such that

Thus $f'$ behaves like a function satisfying the intermediate value theorem , even though $f'$ need not be a continuous map and may have points of discontinuity .