Dini derivative

Remarks

Always $D_+ f(x) \le D^+ f(x)$ and $D_- f(x) \le D^- f(x)$. The right derivative of $f$ at $x$ exists and is finite exactly when $D_+ f(x) = D^+ f(x)$ is finite, and similarly on the left. So $f$ is differentiable at $x$ exactly when all four Dini derivatives are equal and finite.

Example: let $f(x) = x \sin(1/x)$ for $x \ne 0$ and $f(0) = 0$. For $h \ne 0$, the difference quotient at $0$ is $\frac{h \sin(1/h)}{h} = \sin(1/h)$. For $h > 0$ and for $h < 0$ alike, it takes every value in $[-1, 1]$ for arbitrarily small $\abs{h}$. So $D^+ f(0) = D^- f(0) = 1$ and $D_+ f(0) = D_- f(0) = -1$, and $f$ is not differentiable at $0$.

Non-example of finiteness: for $f(x) = \sqrt{\abs{x}}$ at $0$, the quotient for $h > 0$ is $h^{-1/2}$, so $D^+ f(0) = D_+ f(0) = +\infty$. The Dini derivatives exist but are not real numbers.