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
:<math>f:{\mathbb R} \rightarrow {\mathbb R},</math>
is denoted by and defined by
:<math>f'_+(t) = \limsup_{h \to {0+ \frac{f(t + h) - f(t)}{h},</math>
where is the supremum limit and the limit is a one-sided limit. The lower Dini derivative, , is defined by
:<math>f'_-(t) = \liminf_{h \to {0+ \frac{f(t) - f(t - h)}{h},</math>
where is the infimum limit.
If is defined on a vector space, then the upper Dini derivative at in the direction is defined by
:<math>f'_+ (t,d) = \limsup_{h \to {0+ \frac{f(t + hd) - f(t)}{h}.</math>
If is locally Lipschitz, 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 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 .