proof of Darboux’s theorem

Without loss of generality we migth and shall assume f+(a)>t>f-(b). Let g(x):=f(x)-tx. Then g(x)=f(x)-t, g+(a)>0>g-(b), and we wish to find a zero of g.

Since g is a continuous functionMathworldPlanetmathPlanetmath on [a,b], it attains a maximum on [a,b]. Since g+(a)>0 and g+(b)<0 Fermat’s theorem ( states that neither a nor b can be points where f has a local maximumMathworldPlanetmath. So a maximum is attained at some c(a,b). But then g(c)=0 again by Fermat’s theorem (

Title proof of Darboux’s theorem
Canonical name ProofOfDarbouxsTheorem
Date of creation 2013-03-22 12:45:04
Last modified on 2013-03-22 12:45:04
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 7
Author paolini (1187)
Entry type Proof
Classification msc 26A06