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, for which he defined the so-called Dini derivatives.
is denoted by f and 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
- 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).
- 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
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 .
- Lukashenko, T.P. (2001) , "Dini derivative", Encyclopedia of Mathematics, EMS Press.
- Royden, H. L. (1968). Real Analysis (2nd ed.). MacMillan. ISBN 978-0-02-404150-0.
- 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. ISBN 978-1-4348-4161-2.