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

Darboux’ theorem says that, if $f:\mathbb{R}\to \mathbb{R}$ has an antiderivative, than $f$ has to satisfy the *intermediate value property*, namely, for any $$, for any number $C$ with $$ or $$, 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

$$f(x)=\{\begin{array}{ccc}\hfill \frac{1}{x}\mathrm{cos}(\mathrm{ln}x)\hfill & \hfill \text{if}\hfill & \hfill x>0\\ \hfill 0\hfill & \hfill \text{if}\hfill & \hfill x\le 0\end{array}.$$ |

First let us see that $f$ satisfies the intermediate value property. Let $$. If $$ or $b\le 0$, the property is satisfied, since $f$ is continuous^{} on $(-\mathrm{\infty},0]$ and $(0,\mathrm{\infty})$. If $$, we have $f(a)=0$ and $f(b)=(1/b)\mathrm{cos}(\mathrm{ln}b)$. Let $C$ be between $f(a)$ and $(b)$. Let ${a}_{0}=\mathrm{exp}(-2\pi {k}_{0}+\pi )$ for some ${k}_{0}$ large enough such that $$. 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 $G(x)=\mathrm{sin}(\mathrm{ln}x)$ which is defined on $(0,\mathrm{\infty})$.
We have ${G}^{\prime}(x)=f(x)$ on $(0,\mathrm{\infty})$, and since it is a an open connected set, we must have $F(x)=G(x)+c$ on $(0,\mathrm{\infty})$ for some $c\in \mathbb{R}$. But then, we have

$\underset{x\to {0}^{+}}{lim\; sup}F(x)$ | $=\underset{x\to {0}^{+}}{lim\; sup}G(x)+c=1+c$ |

and

$\underset{x\to {0}^{+}}{lim\; inf}F(x)$ | $=\underset{x\to {0}^{+}}{lim\; inf}G(x)+c=-1+c$ |

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

Title | converse of Darboux’s theorem (analysis^{}) is not true |
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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 |