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,

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

denoted by $f'_{+},\,$ is defined as

$f'_{+}(t)=\limsup _{h\rightarrow 0^{+}}{\frac {f(t+h)-f(t)}{h}}.$

The lower Dini derivative, $f'_{-},\,$ is defined as

$f'_{-}(t)=\liminf _{h\rightarrow 0^{+}}{\frac {f(t+h)-f(t)}{h}}$

(see lim sup and lim inf). If $f$ is defined on a vector space, then the upper Dini derivative at $t$ in the direction $d$ is denoted

$f'_{+}(t,d)=\limsup _{h\rightarrow 0^{+}}{\frac {f(t+hd)-f(t)}{h}}.$

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

$D^{-}f(t)=\limsup _{h\rightarrow 0^{-}}{\frac {f(t+h)-f(t)}{h}}.$

and

$D_{-}f(t)=\liminf _{h\rightarrow 0^{-}}{\frac {f(t+h)-f(t)}{h}}.$

• So when using the $D$ 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 $+\infty$ or $-\infty$ at times.

See also

• Derivative (generalizations)

References

• Lukashenko, T.P. (2001) [1994], "Dini derivative", Encyclopedia of Mathematics, EMS Press.
• Royden, H.L. (1968), Real analysis (2nd ed.), MacMillan, ISBN 0-02-40150-5 {{citation}}: Check |isbn= value: length (help).

Dini derivative at PlanetMath.