Prove that a differential equation of the shape
with a continuous function
on an interval I ′ {\displaystyle {}I'} has the solution
where G {\displaystyle {}G} is an antiderivative of g {\displaystyle {}g} such that G ( I ′ ) ⊆ R + {\displaystyle {}G(I')\subseteq \mathbb {R} _{+}} .