The four Dini derivatives of a real function at a point

Let $I\subseteq \mathbb{R}$ be an interval, let $f:I\to \mathbb{R}$, and let $x\in I$.

If $x$ has points of $I$ arbitrarily close on the right, the upper right Dini derivative and lower right Dini derivative of $f$ at $x$ are

$D^{+}f(x):={\mathrm{lim\hspace{0.17em}sup}}_{h\to 0^{+}}\frac{f(x+h)-f(x)}{h},$

$D_{+}f(x):={\mathrm{lim\hspace{0.17em}inf}}_{h\to 0^{+}}\frac{f(x+h)-f(x)}{h}\mathrm{.}$

If $x$ has points of $I$ arbitrarily close on the left, the upper left Dini derivative and lower left Dini derivative of $f$ at $x$ are

$D^{-}f(x):={\mathrm{lim\hspace{0.17em}sup}}_{h\to 0^{+}}\frac{f(x)-f(x-h)}{h},$

$D_{-}f(x):={\mathrm{lim\hspace{0.17em}inf}}_{h\to 0^{+}}\frac{f(x)-f(x-h)}{h}\mathrm{.}$

Each value is taken in the extended real line The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined, so the symbols remain meaningful even when the difference quotients are unbounded. When $x$ is an interior point of $I$ and the usual finite derivative of The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set exists, it is the common finite value of all four Dini derivatives.

• The point of the Dini package is that no boundedness hypothesis is hidden in the notation: $+\mathrm{\infty }$ and $-\mathrm{\infty }$ are permitted values.
• The one-sided direction of the limit is the same $h\to 0^{+}$ in all four formulas. The left derivatives use the quotient at $x-h$, not a separate limit with $h\to 0^{-}$.