Darboux’s theorem (analysis)

Let f:[a,b] be a real-valued continuous functionMathworldPlanetmathPlanetmath on [a,b], which is differentiableMathworldPlanetmathPlanetmath on (a,b), differentiable from the right at a, and differentiable from the left at b. Then f the intermediate value theorem: for every t between f+(a) and f-(b), there is some x[a,b] such that f(x)=t.

Note that when f is continuously differentiable (fC1([a,b])), this is trivially true by the intermediate value theorem. But even when f is not continuous, Darboux’s theorem places a severe restrictionPlanetmathPlanetmath on what it can be.

Title Darboux’s theorem (analysisMathworldPlanetmath)
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 derivativePlanetmathPlanetmath