Real function/Derivative function/Definition

Let ${\displaystyle {}I\subseteq \mathbb {R} }$ denote an interval and let
${\displaystyle f\colon I\longrightarrow \mathbb {R} }$
be a function. We say that ${\displaystyle {}f}$ is differentiable if for every point ${\displaystyle {}a\in I}$ the derivative ${\displaystyle {}f'(a)}$ of ${\displaystyle {}f}$ in ${\displaystyle {}a}$ exists. In this case the mapping
${\displaystyle f'\colon I\longrightarrow \mathbb {R} ,x\longmapsto f'(x),}$
is called the derivative of ${\displaystyle {}f}$.