Right derivative and left derivative

Let $f:E\to\mathbb{R}$ (or $\mathbb{C}$ ) with $E\subseteq\mathbb{R}$ , and let $a\in E$ . If $a$ is a limit point of $E\cap(a,\infty)$ , the right derivative of $f$ at $a$ is

f'_+(a):=\lim_{h\downarrow 0}\frac{f(a+h)-f(a)}{h},

provided the limit exists as a finite number; the limit is taken only over increments with $a+h\in E$ . If $a$ is a limit point of $E\cap(-\infty,a)$ , the left derivative is

f'_-(a):=\lim_{h\uparrow 0}\frac{f(a+h)-f(a)}{h},

provided the limit exists as a finite number; the limit is taken only over increments with $a+h\in E$ .

Relation to differentiability

If both one-sided derivatives exist and are equal, then $f$ is differentiable at $a$ and $f'(a)=f'_+(a)=f'_-(a)$ .

Examples

For $f(x)=|x|$ , one has $f'_+(0)=1$ and $f'_-(0)=-1$ , so $f'(0)$ does not exist.
For $f(x)=x^2$ , $f'_+(a)=f'_-(a)=2a$ for all $a$ .
For the step function $\mathbf{1}_{[0,\infty)}$ , the right derivative at $0$ is $0$ . From the left, the difference quotient is $1/h\to-\infty$ , so no finite left derivative exists.