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# 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 $t$.

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47G30*no label found*

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Oct 19

new correction: examples and OEIS sequences by fizzie

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new correction: Closure properties on languages: DCFL not closed under reversal by babou

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new question: Latent variable by adam_reith

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

new question: Lorenz system by David Bankom

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

Sep 26

new question: Latent variable by adam_reith

## 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 ✓