Dini derivative

The function is said to be pseudoconvex if it is increasing in any direction where the upper Dini derivative is positive.

In mathematics, the Denjoy-Young-Saks theorem gives some possibilities for the Dini derivatives of a function that hold almost everywhere.

In mathematics and, specifically, real analysis, the Dini derivatives (or Dini derivates) are a class of generalizations of the derivative.

If is differentiable at , then the Dini derivative at is the usual derivative at .

If f is a real valued function defined on an interval, then outside a set of measure 0 the Dini derivatives of f satisfy one of the following four conditions at each point:

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.

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).