## You are here

HomeDini derivative

## Primary tabs

# Dini derivative

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

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

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

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

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

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

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 $t$.

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

47G30*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Apr 15

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier

## Corrections

classification by bbukh ✘

D_ by alozano ✘

Vector space by Koro ✓

minor by matte ✓

upper Dini derivative denoted by $f'_+$ by sangwal77 ✓

D_ by alozano ✘

Vector space by Koro ✓

minor by matte ✓

upper Dini derivative denoted by $f'_+$ by sangwal77 ✓