# Darboux’s theorem (analysis)

Let $f:[a,b]\to\mathbb{R}$ be a real-valued continuous function on $[a,b]$, which is differentiable on $(a,b)$, differentiable from the right at $a$, and differentiable from the left at $b$. Then $f^{\prime}$ the intermediate value theorem: for every $t$ between $f^{\prime}_{+}(a)$ and $f^{\prime}_{-}(b)$, there is some $x\in[a,b]$ such that $f^{\prime}(x)=t$.

Note that when $f$ is continuously differentiable ($f\in C^{1}([a,b])$), this is trivially true by the intermediate value theorem. But even when $f^{\prime}$ is not continuous, Darboux’s theorem places a severe restriction on what it can be.

Title Darboux’s theorem (analysis) DarbouxsTheoremanalysis 2013-03-22 12:45:01 2013-03-22 12:45:01 mathwizard (128) mathwizard (128) 7 mathwizard (128) Theorem msc 26A06 intermediate value property of the derivative