# 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*.

## Remarks

- 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. - 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).

## References

- 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.

*This article incorporates material from Dini derivative on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*