Dini derivative

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

$$f_{+}^{\prime }(t)=\underset{h\to 0^{+}}{lim}sup\frac{f(t+h)-f(t)}{h}.$$

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

$$f_{-}^{\prime }(t)=\underset{h\to 0^{+}}{lim}inf\frac{f(t+h)-f(t)}{h}.$$

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

$$f_{+}^{\prime }(t,d)=\underset{h\to 0^{+}}{lim}sup\frac{f(t+hd)-f(t)}{h}.$$

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 $t$.