# 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, for any number $C$ with $f(a) or $f(b), 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. If $0 or $b\leq 0$, the property is satisfied, since $f$ is continuous on $(-\infty,0]$ and $(0,\infty)$. If $a\leq 0, 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}. 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 ConverseOfDarbouxsTheoremanalysisIsNotTrue 2013-03-22 17:33:51 2013-03-22 17:33:51 Gorkem (3644) Gorkem (3644) 6 Gorkem (3644) Example msc 26A06