# Real exponential function/Characterization with derivative/Exercise

${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto f(x),}$
${\displaystyle f'=f{\text{ and }}f(0)=1.}$
Prove that ${\displaystyle {}f(x)=\exp x}$ for all
${\displaystyle {}a\in \mathbb {R} }$.