Peano existence theorem

Statement

Consider a first-order differential equation in explicit form (note that any first-order first-degree differential equation can be converted to such a form):

${\frac {dy}{dx}}=G(x,y)$

Suppose $G$ is a continuous function (in a jointly continuous sense) on an open subset of $\mathbb {R} ^{2}$ containing a point $(x_{0},y_{0})$. Then, there exists a function $f$ defined on an open subset $I$ of $\mathbb {R}$ containing $x_{0}$ such that $f$ satisfies the initial value problem, namely:

$f(x_{0})=y_{0}\qquad {\mbox{ and }}f'(x)=G(x,f(x))\ \forall \ x\in I$