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 ${\displaystyle {}f}$.