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
:
is denoted by and defined by
:
where is the
supremum limit and the limit is a
one-sided limit. The lower Dini derivative, , is defined by
:
where is the
infimum limit.
If is defined on a
vector space, then the upper Dini derivative at in the direction is defined by
:
If is
locally Lipschitz Lipschitz, Lipshitz, or Lipchitz, is an Ashkenazi Jewish (Yiddish/German-Jewish) surname. The surname has many variants, including: Lifshitz ( Lifschitz), Lifshits, Lifshuts, Lefschetz; Lipschitz, Lipshitz, Lipshits, Lopshits, Lipschutz (Lip ...
, 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 mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
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,
:
and
:
.
* 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
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
limit.
* There are two further Dini derivatives, defined to be
:
and
:
.
which are the same as the first pair, but with the
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
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 is differentiable in the usual sense at the point .
* On the
extended reals
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra o ...
, 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
* .
*
*
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Generalizations of the derivative
Real analysis