Dini derivative


In mathematics and, specifically, real analysis, the Dini derivatives are a class of generalizations of the derivative. They were introduced by Ulisse Dini who studied continuous but nondifferentiable functions, for which he defined the so-called Dini derivatives.
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, then is finite. If is differentiable at, then the Dini derivative at is the usual derivative at.

Remarks

and