# Semi-differentiability

## One-dimensional case

Let $latx f$ denote a real-valued function defined on a subset $I$ of the real numbers $\mathbf{R}$. If $a\in I$ is a limit point of $I\cap [a,+\infty)$ and the one-side limit

$\displaystyle f_+(a):=\lim_{x\to a+:x\in I}\frac{f(x)-f(a)}{x-a}$

exists as a real number, then $f$ is called right differentiable at $a$ and the limt $f_+(a)$ is called the right derivative of $f$ at $a$.

Similarily, if $a\in I$ is a limit point of $I\cap (-\infty,a]$ and the one-side limit

$\displaystyle f_-(a):=\lim_{x\to a-,x\in }\frac{f(x)-f(a)}{x-a}$

exists as a real number, then $f$ is called left derivative at $a$ and the limit $f_-(a)$ is called the left derivative of $f$ at $a$.

If $a\in I$ is a limit point of $I\cap [a,+\infty)$ and $I\cap (-\infty,a]$ and if $f$ is left and right differentiable at a, then $f$ is called semi-differentiable at a.