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{{Short description|Class of generalisations of the derivative}}
In [[mathematics]] and, specifically, [[real analysis]], the '''Dini derivatives''' (or '''Dini derivates''') are a class of generalizations of the [[derivative]]. They were introduced by [[Ulisse Dini]], who studied continuous but nondifferentiable functions.

The '''upper Dini derivative''', which is also called an '''upper right-hand derivative''',<ref name="Khalil02">{{cite book | last = Khalil | first = Hassan K. | year = 2002 | edition = 3rd | url = http://www.egr.msu.edu/~khalil/NonlinearSystems/ | isbn = 0-13-067389-7 | title = Nonlinear Systems | publisher = [[Prentice Hall]] | location = Upper Saddle River, NJ}}</ref> of a [[continuous function]]

:<math>f:{\mathbb R} \rightarrow {\mathbb R},</math>

is denoted by {{math|''f''{{underset|+|′}}}} and defined by

:<math>f'_+(t) = \limsup_{h \to {0+}} \frac{f(t + h) - f(t)}{h},</math>

where {{math|lim sup}} is the [[supremum limit]] and the limit is a [[one-sided limit]]. The '''lower Dini derivative''', {{math|''f''{{underset|−|′}}}}, is defined by

:<math>f'_-(t) = \liminf_{h \to {0+}} \frac{f(t) - f(t - h)}{h},</math>

where {{math|lim inf}} is the [[infimum limit]].

If {{math|''f''}} is defined on a [[vector space]], then the upper Dini derivative at {{math|''t''}} in the direction {{math|''d''}} is defined by

:<math>f'_+ (t,d) = \limsup_{h \to {0+}} \frac{f(t + hd) - f(t)}{h}.</math>

If {{math|''f''}} is [[locally]] [[Lipschitz continuity|Lipschitz]], then {{math|''f''{{underset|+|′}}}} is finite. If {{math|''f''}} is [[differentiable function|differentiable]] at {{math|''t''}}, then the Dini derivative at {{math|''t''}} is the usual [[derivative]] at {{math|''t''}}.

==Remarks==
* The functions are defined in terms of the [[infimum]] and [[supremum]] in order to make the Dini derivatives as "bullet proof" as possible, so that the Dini derivatives are well-defined for almost all functions, even for functions that are not conventionally differentiable. The upshot of Dini's analysis is that a function is differentiable at the point {{mvar|t}} on the real line ({{math|ℝ}}), only if all the Dini derivatives exist, and have the same value.

* Sometimes the notation {{math|''D''<sup>+</sup> ''f''(''t'')}} is used instead of {{math|''f''{{underset|+|′}}(''t'')}} and {{math|''D''<sub>−</sub> ''f''(''t'')}} is used instead of {{math|''f''{{underset|−|′}}(''t'')}}.<ref name="Khalil02"/>
* Also,
:<math>D^{+}f(t) = \limsup_{h \to {0+}} \frac{f(t + h) - f(t)}{h}</math>

and

:<math>D_{-}f(t) = \liminf_{h \to {0+}} \frac{f(t) - f(t - h)}{h}</math>.

* So when using the {{math|''D''}} notation of the Dini derivatives, the plus or minus sign indicates the left- or right-hand limit, and the placement of the sign indicates the [[infimum]] or [[supremum]] limit.

* There are two further Dini derivatives, defined to be

:<math>D_{+}f(t) = \liminf_{h \to {0+}} \frac{f(t + h) - f(t)}{h}</math>

and

:<math>D^{-}f(t) = \limsup_{h \to {0+}} \frac{f(t) - f(t - h)}{h}</math>.

which are the same as the first pair, but with the [[supremum]] and the [[infimum]] reversed. For only moderately ill-behaved functions, the two extra Dini derivatives aren't needed. For particularly badly behaved functions, if all four Dini derivatives have the same value (<math>D^{+}f(t) = D_{+}f(t) = D^{-}f(t) = D_{-}f(t)</math>) then the function {{mvar|f}} is differentiable in the usual sense at the point {{mvar|t}}&nbsp;.

* On the [[extended real number line|extended reals]], each of the Dini derivatives always exist; however, they may take on the values {{math|+∞}} or {{math|−∞}} at times (i.e., the Dini derivatives always exist in the [[extended real number line|extended]] sense).

==See also==

* {{annotated link|Denjoy–Young–Saks theorem}}
* {{annotated link|Derivative (generalizations)}}
* {{annotated link|Semi-differentiability}}

==References==
{{reflist|25em}}
{{refbegin}}
* {{springer|id=d/d032530|title=Dini derivative|first=T.P.|last=Lukashenko|year=2001}}.
* {{Cite book |first=H. L. |last=Royden |title=Real Analysis |publisher=MacMillan |year=1968 |edition=2nd |isbn=978-0-02-404150-0}}
* {{cite book|first1=Brian S. |last1=Thomson|first2=Judith B. |last2=Bruckner|first3=Andrew M. |last3=Bruckner|title=Elementary Real Analysis|year=2008|publisher=ClassicalRealAnalysis.com [first edition published by Prentice Hall in 2001]|isbn=978-1-4348-4161-2|pages=301–302}}
{{refend}}

{{PlanetMath attribution|id=4714|title=Dini derivative}}

[[Category:Generalizations of the derivative]]
[[Category:Real analysis]]