Dini derivative

In mathematics and, specifically, real analysis, the Dini derivatives (or Dini derivates) are a class of generalizations of the derivative. The upper Dini derivative, which is also called an upper right-hand derivative, 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 limsup  is the supremum limit. The lower Dini derivative, $f'_-,\,$, is defined by

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

where liminf  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 $f'_-(t).\,$

• Also,

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

and

$D_-f(t) \triangleq \liminf_{h \to {0-}} \frac{f(t + h) - f(t)}{h}.$

• 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).

See also

• Derivative (generalizations)

References

In-line references

General references

• Lukashenko, T.P. (2001), "Dini derivative", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=d/d032530 .
• Royden, H.L. (1968). Real analysis (2nd ed.). MacMillan. ISBN 0-02-40150-5 .

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