Peano existence theorem

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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