Dini 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, of a continuous function

:$f: \rightarrow ,$

is denoted by and defined by

:$f'_+(t) = \limsup_ \frac,$

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

:$f'_-(t) = \liminf_ \frac,$

where is the infimum limit.

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

:$f'_+ (t,d) = \limsup_ \frac.$

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

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 on the real line (), only if all the Dini derivatives exist, and have the same value.

* Sometimes the notation is used instead of and is used instead of .
* Also,
:$D^f(t) = \limsup_ \frac$

and

:$D_f(t) = \liminf_ \frac$.

* So when using the 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

:$D_f(t) = \liminf_ \frac$

and

:$D^f(t) = \limsup_ \frac$.

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 is differentiable in the usual sense at the point  .

* 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

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References

* .
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Generalizations of the derivative
Real analysis