View source for Dini derivative

In the [[mathematics]], and specifically [[real analysis]], the '''Dini derivatives''' (or '''Dini derivates''') are a class of generalizations of the [[derivative]].  The '''upper Dini derivative''' of a [[continuous function]], 

:<math>f:{\mathbb R} \rightarrow {\mathbb R},</math> 

denoted by  <math>f'_+,\,</math> is defined as

:<math>f'_+(t) = \limsup_{h\rightarrow 0^+} \frac{f(t + h) - f(t)}{h}.</math>

The '''lower Dini derivative''', <math>f'_-,\,</math> is defined as

:<math>f'_-(t) = \liminf_{h\rightarrow 0^+} \frac{f(t + h) - f(t)}{h}</math>

(see [[limit superior and limit inferior|lim sup and lim inf]]). If <math>f</math> is defined on a [[vector space]], then the upper Dini derivative at <math>t</math> in the direction <math>d</math> is denoted

:<math>f'_+ (t,d) = \limsup_{h\rightarrow 0^+} \frac{f(t + hd) - f(t)}{h}.</math>

If <math>f</math> is [[locally]] [[Lipschitz continuity|Lipschitz]] then <math>f'_+\,</math> is finite. If <math>f</math> is [[differentiable function|differentiable]] at <math>t</math>, then the Dini derivative at <math>t</math> is the usual [[derivative]] at <math>t.</math>

==Remarks==
* Sometimes the notation <math>D^+ f(t)\,</math> is used instead of <math>f'_+(t),\,</math> and <math>D_+f(t)\,</math> is used instead of <math>f'_-(t).\,</math>

* Also 
:<math>D^-f(t) = \limsup_{h\rightarrow 0^-} \frac{f(t + h) - f(t)}{h}.</math>

and

:<math>D_-f(t) = \liminf_{h\rightarrow 0^-} \frac{f(t + h) - f(t)}{h}.</math>

* So when using the <math> D </math> notation of the Dini Derivates, the plus or minus sign indicates the left or right hand limits, and the placement of the sign indicates the liminf or limsup.

* Each of the Dini derivatives always exist; however, they may take on the values <math>+ \infty</math> or <math>- \infty</math> at times.

==See also==

* [[Derivative (generalizations)]]

==References==
* {{springer|id=d/d032530|title=Dini derivative|first=T.P.|last=Lukashenko|year=2001}}.
* {{citation|first=H.L.|last=Royden|title=Real analysis|publisher=MacMillan|year=1968|edition=2nd|isbn=0-02-40150-5}}.

{{planetmath|title=Dini derivative|id=4714}}
[[Category:Generalizations of the derivative]]
[[Category:Real analysis]]

[[hu:Dini-derivált]]