Dini derivative

In mathematics and, specifically, real analysis, the Dini derivatives (or Dini derivates) are a class of generalizations of the derivative. They were introduced by Ulisse Dini, who studied continuous but nondifferentiable functions.

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


 * $$f'_+(t) = \limsup_{h \to {0+}} \frac{f(t + h) - f(t)}{h},$$

where $lim sup$ is the supremum limit and the limit is a one-sided limit. The lower Dini derivative, $f′ −$, is defined by


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

where $lim inf$ 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


 * $$f'_+ (t,d) = \limsup_{h \to {0+}} \frac{f(t + hd) - f(t)}{h}.$$

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

 * The functions are defined in terms of the infimum and supremum in order to make the Dini derivatives as "bullet proof" as possible, so that the Dini derivatives are well-defined for almost all functions, even for functions that are not conventionally differentiable. The upshot of Dini's analysis is that a function is differentiable at the point $t$ on the real line ($ℝ$), only if all the Dini derivatives exist, and have the same value.


 * Sometimes the notation $D^{+} f(t)$ is used instead of $f′ +(t)$ and $D_{−} f(t)$ is used instead of $f′ −(t)$.
 * Also,
 * $$D^{+}f(t) = \limsup_{h \to {0+}} \frac{f(t + h) - f(t)}{h}$$

and


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


 * 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.


 * There are two further Dini derivatives, defined to be


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

and


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

which are the same as the first pair, but with the supremum and the infimum reversed. For only moderately ill-behaved functions, the two extra Dini derivatives aren't needed. For particularly badly behaved functions, if all four Dini derivatives have the same value ($$D^{+}f(t) = D_{+}f(t) = D^{-}f(t) = D_{-}f(t)$$) then the function $f$ is differentiable in the usual sense at the point $t$.


 * On the extended reals, each of the Dini derivatives always exist; however, they may take on the values $+∞$ or $−∞$ at times (i.e., the Dini derivatives always exist in the extended sense).