Darboux’s theorem (analysis)

Let $f:[a,b]\to ℝ$ 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)
Canonical name	DarbouxsTheoremanalysis
Date of creation	2013-03-22 12:45:01
Last modified on	2013-03-22 12:45:01
Owner	mathwizard (128)
Last modified by	mathwizard (128)
Numerical id	7
Author	mathwizard (128)
Entry type	Theorem
Classification	msc 26A06
Synonym	intermediate value property of the derivative