Dini derivative

Template: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, of a continuous function

$f:ℝ\to ℝ,$

is denoted by Template:Math and defined by

$f{\text{'}}_{+}(t)=\underset{h\to 0+}{\mathrm{lim\; sup}}\frac{f(t+h)-f(t)}{h},$

where Template:Math is the supremum limit and the limit is a one-sided limit. The lower Dini derivative, Template:Math, is defined by

$f{\text{'}}_{-}(t)=\underset{h\to 0+}{\mathrm{lim\; inf}}\frac{f(t)-f(t-h)}{h},$

where Template:Math is the infimum limit.

If Template:Math is defined on a vector space, then the upper Dini derivative at Template:Math in the direction Template:Math is defined by

$f{\text{'}}_{+}(t,d)=\underset{h\to 0+}{\mathrm{lim\; sup}}\frac{f(t+hd)-f(t)}{h}.$

If Template:Math is locally Lipschitz, then Template:Math is finite. If Template:Math is differentiable at Template:Math, then the Dini derivative at Template:Math is the usual derivative at Template: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 Template:Mvar on the real line (Template:Math), only if all the Dini derivatives exist, and have the same value.

• Sometimes the notation Template:Math is used instead of Template:Math and Template:Math is used instead of Template:Math.
• Also,

$D^{+}f(t)=\underset{h\to 0+}{\mathrm{lim\; sup}}\frac{f(t+h)-f(t)}{h}$

and

$D_{-}f(t)=\underset{h\to 0+}{\mathrm{lim\; inf}}\frac{f(t)-f(t-h)}{h}$.

• So when using the Template: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

$D_{+}f(t)=\underset{h\to 0+}{\mathrm{lim\; inf}}\frac{f(t+h)-f(t)}{h}$

and

$D^{-}f(t)=\underset{h\to 0+}{\mathrm{lim\; sup}}\frac{f(t)-f(t-h)}{h}$.

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 ($D^{+}f(t)=D_{+}f(t)=D^{-}f(t)=D_{-}f(t)$) then the function Template:Mvar is differentiable in the usual sense at the point Template:Mvar .

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

See also

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References

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