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 $\R^2$ containing a point $(x_0,y_0)$. Then, there exists a function $f$ defined on an open subset $I$ of $\R$ containing $x_0$ such that $f$ satisfies the initial value problem, namely:

$f(x_0) = y_0 \qquad \mbox{ and } f'(x) = G(x,f(x)) \ \forall \ x \in I$