# Dini derivative

The upper Dini derivative^{} of a continuous function^{}, $f:\mathbf{R}\mapsto \mathbf{R}$, denoted by ${f}_{+}^{\prime}$, is defined as

$${f}_{+}^{\prime}(t)=\underset{h\to {0}^{+}}{lim}sup\frac{f(t+h)-f(t)}{h}.$$ |

The lower Dini derivative, ${f}_{-}^{\prime}$, is defined as

$${f}_{-}^{\prime}(t)=\underset{h\to {0}^{+}}{lim}inf\frac{f(t+h)-f(t)}{h}.$$ |

Remark: Sometimes the notation ${D}^{+}f(t)$ is used instead of ${f}_{+}^{\prime}(t)$, and ${D}^{-}f(t)$ is used instead of ${f}_{-}^{\prime}(t)$.

Remark: Like conventional derivatives, Dini derivatives do not always exist.

If $f$ is defined on a vector space, then the upper Dini derivative at $t$ in the direction $d$ is denoted

$${f}_{+}^{\prime}(t,d)=\underset{h\to {0}^{+}}{lim}sup\frac{f(t+hd)-f(t)}{h}.$$ |

If $f$ is locally Lipschitz^{} then ${D}^{+}f$ is finite. If $f$ is differentiable^{} at $t$ then the Dini derivative at $t$ is the derivative at $t$.

Title | Dini derivative |
---|---|

Canonical name | DiniDerivative |

Date of creation | 2013-03-22 13:57:00 |

Last modified on | 2013-03-22 13:57:00 |

Owner | lha (3057) |

Last modified by | lha (3057) |

Numerical id | 11 |

Author | lha (3057) |

Entry type | Definition |

Classification | msc 47G30 |