# Dini derivative

The upper Dini derivative of a continuous function, $f:{\bf R}\mapsto{\bf R}$, denoted by $f^{\prime}_{+}$, is defined as

 $f^{\prime}_{+}(t)=\lim_{h\rightarrow 0^{+}}\sup\frac{f(t+h)-f(t)}{h}.$

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

 $f^{\prime}_{-}(t)=\lim_{h\rightarrow 0^{+}}\inf\frac{f(t+h)-f(t)}{h}.$

Remark: Sometimes the notation $D^{+}f(t)$ is used instead of $f^{\prime}_{+}(t)$, and $D^{-}f(t)$ is used instead of $f^{\prime}_{-}(t)$.

Remark: Like conventional derivatives, Dini derivatives do not always exist.

If $f$ is defined on a vector space, then the upper Dini derivative at $t$ in the direction $d$ is denoted

 $f^{\prime}_{+}(t,d)=\lim_{h\rightarrow 0^{+}}\sup\frac{f(t+hd)-f(t)}{h}.$

If $f$ is locally Lipschitz then $D^{+}f$ is finite. If $f$ is differentiable at $t$ then the Dini derivative at $t$ is the derivative at $t$.

Title Dini derivative DiniDerivative 2013-03-22 13:57:00 2013-03-22 13:57:00 lha (3057) lha (3057) 11 lha (3057) Definition msc 47G30