Using the Peano existence theorem (ODE's) to imply the implicit function theorem

Question

The question, found in Hale's Ordinary Differential Equations, is to

State an implicit function theorem whose validity will be implied by the Peano existence theorem.

I am confused on how this should be done, because the Peano existence theorem in the book used the implicit function theorem within the proof. I realize there are other ways to prove Peano existence, such as Arzela-Ascoli, so perhaps this isn't a problem. Nonetheless, I am out of ideas of how this statement should be posed.

Any conceptual intuition or plain answers would be greatly appreciated.

Answer 1

Suppose you are asked to find $y$ as a function of $x$ by solving the implicit function $f(x,y) = 0$. You can attempt to do so by formally writing $y = y(x)$ and taking the total derivative $\frac{d}{dx}$. This gives you an ODE for $y$ in terms of $x$. Apply Peano.

(Note, there are certain assumptions required to apply Peano theorem, translate those back to assumptions on $f$ near some base point $(x_0,y_0)$ satisfying $f(x_0,y_0) = 0$. Compare that to the usual statement of implicit function theorem.)

Answer 2

This answer is taken from the The implicit function theorem:history,theory and applications by Steven G. Krantz and Harold R.Parks: