Definition 2

Let $\mathbf{f}$ be a vector function defined on an interval $I \subset \mathbf{R}$, and let $x_0$ be a point of $I$ such that the interval $I \cap [x_0, +\infty[$ (resp. $I \cap ]-\infty, x_0]$) does not reduce to a single point. We say that $\mathbf{f}$ is differentiable on the right (resp. on the left) at the point $x_0$ if the restriction of $\mathbf{f}$ to the interval $I \cap [x_0, +\infty[$ (resp. $I \cap ]-\infty, x_0]$) is differentiable at the point $x_0$; the value of the derivative of this restriction at the point $x_0$ is called the right (resp. left) derivative of $\mathbf{f}$ at the point $x_0$ and is denoted by $\mathbf{f}'_d(x_0)$ (resp. $\mathbf{f}'_g(x_0)$).

Let $\mathbf{f}$ be a vector function defined on $I$, and $x_0$ an interior point of $I$ such that $\mathbf{f}$ is continuous at this point; it follows from defs. 1 and 2 that for $\mathbf{f}$ to be differentiable at $x_0$ it is necessary and sufficient that $\mathbf{f}$ admit both a right and a left derivative at this point, and that these derivatives be equal; and then