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 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 .

Remarks

ℝ ), only if all the Dini derivatives exist, and have the same value. D+ f(t) is used instead of f′ +(t) and D− f(t) is used instead of f′ −(t) . and D 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. 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. +∞ or −∞ at times (i.e., the Dini derivatives always exist in the extended sense).