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
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.
and
and
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 () then the function f is differentiable in the usual sense at the point t .
This article incorporates material from Dini derivative on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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
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.
and
and
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 () then the function f is differentiable in the usual sense at the point t .
This article incorporates material from Dini derivative on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.