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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]]. The '''upper Dini derivative''', which is also called an '''upper right-hand derivative''',<ref name="Khalil02">{{cite book
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.
| last = Khalil

| first = H.K.
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]]
| authorlink = Hassan K. Khalil
| 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>
:<math>f:{\mathbb R} \rightarrow {\mathbb R},</math>


is denoted by <math>f'_+,\,</math> and defined by
is denoted by {{math|''f''{{underset|+|′}}}} and defined by


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


where <math>\limsup</math> is the [[supremum limit]]. The '''lower Dini derivative''', <math>f'_-,\,</math>, is defined by
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) \triangleq \liminf_{h \to {0+}} \frac{f(t + h) - f(t)}{h}</math>
:<math>f'_-(t) = \liminf_{h \to {0+}} \frac{f(t) - f(t - h)}{h},</math>


where <math>\liminf</math> is the [[infimum limit]].
where {{math|lim inf}} is the [[infimum limit]].


If <math>f</math> is defined on a [[vector space]], then the upper Dini derivative at <math>t</math> in the direction <math>d</math> is defined by
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) \triangleq \limsup_{h \to {0+}} \frac{f(t + hd) - f(t)}{h}.</math>
:<math>f'_+ (t,d) = \limsup_{h \to {0+}} \frac{f(t + hd) - f(t)}{h}.</math>


If <math>f</math> is [[locally]] [[Lipschitz continuity|Lipschitz]], then <math>f'_+\,</math> is finite. If <math>f</math> is [[differentiable function|differentiable]] at <math>t</math>, then the Dini derivative at <math>t</math> is the usual [[derivative]] at <math>t</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==
==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^+ f(t)\,</math> is used instead of <math>f'_+(t),\,</math> and <math>D_+f(t)\,</math> is used instead of <math>f'_-(t).\,</math><ref name="Khalil02"/>


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


and
and


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


* So when using the <math> D </math> 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.
* 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
* On the [[extended real number line|extended reals]], each of the Dini derivatives always exist; however, they may take on the values <math>+ \infty</math> or <math>- \infty</math> at times (i.e., the Dini derivatives always exist in the [[extended real number line|extended]] sense).

:<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==
==See also==


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


==References==
==References==
{{reflist|25em}}
; In-line references
{{reflist}}

; General references
{{refbegin}}
{{refbegin}}
* {{springer|id=d/d032530|title=Dini derivative|first=T.P.|last=Lukashenko|year=2001}}.
* {{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=0-02-40150-5|postscript=<!--None-->}}.
* {{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}}
{{refend}}


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

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

[[fr:Dérivées de Dini]]
[[hu:Dini-derivált]]
[[pl:Pochodna Diniego]]

Latest revision as of 05:08, 24 May 2024

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,[1] of a continuous function

is denoted by f+ and defined by

where lim sup is the supremum limit and the limit is a one-sided limit. The lower Dini derivative, f, is defined by

where lim inf is the infimum limit.

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

If f is locally Lipschitz, then f+ is finite. If f is differentiable at t, then the Dini derivative at t is the usual derivative at t.

Remarks

[edit]
  • 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 t on the real line (), only if all the Dini derivatives exist, and have the same value.
  • Sometimes the notation D+ f(t) is used instead of f+(t) and D f(t) is used instead of f(t).[1]
  • Also,

and

.
  • So when using the 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

and

.

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 () then the function f is differentiable in the usual sense at the point t .

  • On the extended reals, each of the Dini derivatives always exist; however, they may take on the values +∞ or −∞ at times (i.e., the Dini derivatives always exist in the extended sense).

See also

[edit]

References

[edit]
  1. ^ a b Khalil, Hassan K. (2002). Nonlinear Systems (3rd ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-067389-7.

This article incorporates material from Dini derivative on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.