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

f:{\mathbb {R} }\rightarrow {\mathbb {R} },

is denoted by $f' +$ and defined by

f'_{+}(t)\triangleq \limsup _{h\to {0+}}{\frac {f(t+h)-f(t)}{h}},

where $lim sup$ is the supremum limit and the limit is a one-sided limit. The lower Dini derivative, $f' -$ , is defined by

f'_{-}(t)\triangleq \liminf _{h\to {0+}}{\frac {f(t+h)-f(t)}{h}},

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

f'_{+}(t,d)\triangleq \limsup _{h\to {0+}}{\frac {f(t+hd)-f(t)}{h}}.

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,

D^{+}f(t)\triangleq \limsup _{h\to {0+}}{\frac {f(t)-f(t-h)}{h}}

and

D_{-}f(t)\triangleq \liminf _{h\to {0-}}{\frac {f(t)-f(t-h)}{h}}

.

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 $+\infty$ or $-\infty$ at times (i.e., the Dini derivatives always exist in the extended sense).

References

^ ^a ^b 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", 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.

This article incorporates material from Dini derivative on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.^{[failed verification]}

[Khalil02-1] Khalil, Hassan K. (2002). Nonlinear Systems (3rd ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-067389-7.

[1]

Remarks

See also

References