converse of Darboux’s theorem (analysis) is not true

Darboux’ theorem says that, if f: has an antiderivative, than f has to satisfy the intermediate value property, namely, for any a<b, for any number C with f(a)<C<f(b) or f(b)<C<f(a), there exists a c(a,b) such that f(c)=C. With this theorem, we understand that if f does not satisfy the intermediate value property, then no function F satisfies F=f on .

Now, we will give an example to show that the converseMathworldPlanetmath is not true, i.e., a function that satisfies the intermediate value property might still have no antiderivative.



First let us see that f satisfies the intermediate value property. Let a<b. If 0<a or b0, the property is satisfied, since f is continuousMathworldPlanetmathPlanetmath on (-,0] and (0,). If a0<b, we have f(a)=0 and f(b)=(1/b)cos(lnb). Let C be between f(a) and (b). Let a0=exp(-2πk0+π) for some k0 large enough such that a0<b. Then f(a0)=0=f(a), and since f is continuous on (a0,b), we must have a c(a0,b) with f(c)=C.

Assume, for a contradictionMathworldPlanetmathPlanetmath that there exists a differentiable function F such that F(x)=f(x) on . Then consider the function G(x)=sin(lnx) which is defined on (0,). We have G(x)=f(x) on (0,), and since it is a an open connected set, we must have F(x)=G(x)+c on (0,) for some c. But then, we have

lim supx0+F(x) =lim supx0+G(x)+c=1+c


lim infx0+F(x) =lim infx0+G(x)+c=-1+c

which contradicts the differentiability of F at 0.

Title converse of Darboux’s theorem (analysisMathworldPlanetmath) is not true
Canonical name ConverseOfDarbouxsTheoremanalysisIsNotTrue
Date of creation 2013-03-22 17:33:51
Last modified on 2013-03-22 17:33:51
Owner Gorkem (3644)
Last modified by Gorkem (3644)
Numerical id 6
Author Gorkem (3644)
Entry type Example
Classification msc 26A06