Dini derivative: Difference between revisions
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:<math>f:{\mathbb R} \rightarrow {\mathbb R},</math> |
:<math>f:{\mathbb R} \rightarrow {\mathbb R},</math> |
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is denoted by {{math|''f''{{ |
is denoted by {{math|''f''{{underset|+|′}}}} and defined by |
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:<math>f'_+(t) \triangleq \limsup_{h \to {0+}} \frac{f(t + h) - f(t)}{h},</math> |
:<math>f'_+(t) \triangleq \limsup_{h \to {0+}} \frac{f(t + h) - f(t)}{h},</math> |
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where {{math|lim sup}} is the [[supremum limit]] and the limit is a [[one-sided limit]]. The '''lower Dini derivative''', {{math|''f''{{ |
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 |
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:<math>f'_-(t) \triangleq \liminf_{h \to {0+}} \frac{f(t + h) - f(t)}{h},</math> |
:<math>f'_-(t) \triangleq \liminf_{h \to {0+}} \frac{f(t + h) - f(t)}{h},</math> |
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:<math>f'_+ (t,d) \triangleq \limsup_{h \to {0+}} \frac{f(t + hd) - f(t)}{h}.</math> |
:<math>f'_+ (t,d) \triangleq \limsup_{h \to {0+}} \frac{f(t + hd) - f(t)}{h}.</math> |
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If {{math|''f''}} is [[locally]] [[Lipschitz continuity|Lipschitz]], then {{math|''f''{{ |
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''}}. |
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==Remarks== |
==Remarks== |
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* Sometimes the notation {{math|''D''<sup>+</sup>''f''(''t'')}} is used instead of {{math|''f''{{ |
* 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"/> |
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* Also, |
* Also, |
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:<math>D^-f(t) \triangleq \limsup_{h \to {0-}} \frac{f(t + h) - f(t)}{h}</math> |
:<math>D^-f(t) \triangleq \limsup_{h \to {0-}} \frac{f(t + h) - f(t)}{h}</math> |
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Revision as of 19:03, 10 December 2016
 | This article may be too technical for most readers to understand. (May 2016) |
| This article may require cleanup to meet Wikipedia's quality standards. The specific problem is: Clarify there are four of them for a single-variable real function. Clarify how each pair coincides with the left and right derivatives when these exist etc. Also, the function probably doesn't need to be continuous. Add EXAMPLES! (April 2015) |
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.
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
- 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.
- 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
References
- ^ a b Khalil, Hassan K. (2002). Nonlinear Systems (3rd ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-067389-7.
- Lukashenko, T.P. (2001) [1994], "Dini derivative", Encyclopedia of Mathematics, EMS Press.
- Royden, H. L. (1968). Real Analysis (2nd ed.). MacMillan. ISBN 978-0-02-404150-0.
- Thomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M. (2008). Elementary Real Analysis. ClassicalRealAnalysis.com [first edition published by Prentice Hall in 2001]. pp. 301–302. ISBN 978-1-4348-4161-2.
This article incorporates material from Dini derivative on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.[failed verification]