Английская Википедия:Dini derivative
Шаблон:Short description 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
- <math>f:{\mathbb R} \rightarrow {\mathbb R},</math>
is denoted by Шаблон:Math and defined by
- <math>f'_+(t) = \limsup_{h \to {0+}} \frac{f(t + h) - f(t)}{h},</math>
where Шаблон:Math is the supremum limit and the limit is a one-sided limit. The lower Dini derivative, Шаблон:Math, is defined by
- <math>f'_-(t) = \liminf_{h \to {0+}} \frac{f(t) - f(t - h)}{h},</math>
where Шаблон:Math is the infimum limit.
If Шаблон:Math is defined on a vector space, then the upper Dini derivative at Шаблон:Math in the direction Шаблон:Math is defined by
- <math>f'_+ (t,d) = \limsup_{h \to {0+}} \frac{f(t + hd) - f(t)}{h}.</math>
If Шаблон:Math is locally Lipschitz, then Шаблон:Math is finite. If Шаблон:Math is differentiable at Шаблон:Math, then the Dini derivative at Шаблон:Math is the usual derivative at Шаблон:Math.
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 Шаблон:Mvar on the real line (Шаблон:Math), only if all the Dini derivatives exist, and have the same value.
- Sometimes the notation Шаблон:Math is used instead of Шаблон:Math and Шаблон:Math is used instead of Шаблон:Math.[1]
- 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 Шаблон: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.
- 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 Шаблон:Mvar is differentiable in the usual sense at the point Шаблон:Mvar .
- On the extended reals, each of the Dini derivatives always exist; however, they may take on the values Шаблон:Math or Шаблон:Math at times (i.e., the Dini derivatives always exist in the extended sense).
See also
References
Шаблон:Reflist Шаблон:Refbegin
Шаблон:PlanetMath attributionШаблон:Failed verification
