Dini derivative

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.

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

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.


  • Sometimes the notation D+ f(t) is used instead of f+(t) and D f(t) is used instead of f(t).[1]
  • Also,


  • 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.
  • 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).

See also


  1. Khalil, Hassan K. (2002). Nonlinear Systems (3rd ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-067389-7.
  • Lukashenko, T.P. (2001) [1994], "Dini derivative", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4.
  • 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.

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