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,[1] 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
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 t on the real line (ℝ), only if all the Dini derivatives exist, and have the same value.
Sometimes the notation D+f(t) is used instead of f′+(t) and D−f(t) is used instead of f′−(t).[1]
Also,
and
.
So when using the 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.
There are two further Dini derivatives, defined to be
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 .
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).
Thomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M. (2008). Elementary Real Analysis. ClassicalRealAnalysis.com [first edition published by Prentice Hall in 2001]. pp. 301–302.
ISBN978-1-4348-4161-2.