In mathematics and, specifically, real analysis, the Dini derivatives (or Dini derivates) are a class of generalizations of the derivative. 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. 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
- Sometimes the notation is used instead of and is used instead of 
- 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).
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- General references