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

Darboux’ theorem says that, if $f\colon\mathbb{R}\rightarrow\mathbb{R}$ 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 $c\in(a,b)$ 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 $F^{\prime}=f$ on $\mathbb{R}$ .

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

Let

$f(x)=\left\{\begin{array}[]{ccr}\displaystyle\frac{1}{x}\cos(\ln x)&\mbox{if}&% x>0\\ 0&\mbox{if}&x\leq 0\end{array}\right..$

First let us see that $f$ satisfies the intermediate value property. Let $a<b$ . If $0<a$ or $b\leq 0$ , the property is satisfied, since $f$ is continuous on $(-\infty,0]$ and $(0,\infty)$ . If $a\leq 0<b$ , we have $f(a)=0$ and $f(b)=(1/b)\cos(\ln b)$ . Let $C$ be between $f(a)$ and $(b)$ . Let $a_{0}=\exp(-2\pi k_{0}+\pi)$ for some $k_{0}$ large enough such that $a_{0}<b$ . Then $f(a_{0})=0=f(a)$ , and since $f$ is continuous on $(a_{0},b)$ , we must have a $c\in(a_{0},b)$ with $f(c)=C$ .

Assume, for a contradiction that there exists a differentiable function $F$ such that $F^{\prime}(x)=f(x)$ on $\mathbb{R}$ . Then consider the function $G(x)=\sin(\ln x)$ which is defined on $(0,\infty)$ . We have $G^{\prime}(x)=f(x)$ on $(0,\infty)$ , and since it is a an open connected set, we must have $F(x)=G(x)+c$ on $(0,\infty)$ for some $c\in\mathbb{R}$ . But then, we have

$\displaystyle\limsup_{x\rightarrow 0^{+}}F(x)$

$\displaystyle=\limsup_{x\rightarrow 0^{+}}G(x)+c=1+c$

and

$\displaystyle\liminf_{x\rightarrow 0^{+}}F(x)$

$\displaystyle=\liminf_{x\rightarrow 0^{+}}G(x)+c=-1+c$

which contradicts the differentiability of $F$ at $0$ .

Title	converse of Darboux’s theorem (analysis) 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