# Darboux’s theorem (analysis)

Let $f:[a,b]\to \mathbb{R}$ be a real-valued continuous function^{} on $[a,b]$, which is differentiable^{} on $(a,b)$, differentiable from the right at $a$, and differentiable from the left at $b$. Then ${f}^{\prime}$ the intermediate value theorem: for every $t$ between ${f}_{+}^{\prime}(a)$ and ${f}_{-}^{\prime}(b)$, there is some $x\in [a,b]$ such that ${f}^{\prime}(x)=t$.

Note that when $f$ is continuously differentiable ($f\in {C}^{1}([a,b])$), this is trivially true *by* the intermediate value theorem. But even when ${f}^{\prime}$ is *not* continuous, Darboux’s theorem places a severe restriction^{} on what it can be.

Title | Darboux’s theorem (analysis^{}) |
---|---|

Canonical name | DarbouxsTheoremanalysis |

Date of creation | 2013-03-22 12:45:01 |

Last modified on | 2013-03-22 12:45:01 |

Owner | mathwizard (128) |

Last modified by | mathwizard (128) |

Numerical id | 7 |

Author | mathwizard (128) |

Entry type | Theorem |

Classification | msc 26A06 |

Synonym | intermediate value property of the derivative^{} |