Imagine you are given $f(x,y) := y^2-\sin(x)^2$ and you want to answer the question, if there is a neighbourhood of $x=0$ such that $f(x,y(x))=0$ with $y(0)=0$.

One idea that comes to mind is the implicit function theorem: We readily verify that $f(0,0)=0$. Also $f$ is smooth in both its elements.

However, $\partial_2 f(0,0)=0,$ which means that the assumptions of the implicit function theorem are not met.

This is clear since $y = \sin(x)$ and $y=-\sin(x)$ are both solutions. Therefore, there is no unique solution in the neighbourhood of zero.

For our particular $f$ there is of course no need to invoke any heavy tools, since everything is explicit. However, I wonder if for functions of this type there is a different abstract method to argue that there exists at least one solution $y(x)$ in a neighbourhood of $x=0$ (disregarding the uniqueness).

somethingmore than just $f(0,0) = 0$ to get existence. $\endgroup$3more comments