# Implicit Function Theorem for $y=x^3$

In a simple case of the IFT, consider $$f(x,y)=y-x^3:\mathbb{R}^2\to\mathbb{R}$$ Now we have that $f(0,0)=0$ and $\frac{\partial f}{\partial x}(0,0)=-3(0)^2=0$, so it would seem that we can't solve for $x(y)$ in a neighborhood of the origin.

That seems counter-intuitive, however, since we can surely write $x=\sqrt[3]{y}$, which seems to capture the essence of the function exactly, even near/at the origin.

I'm sure I am making a silly mistake, but I'd very much appreciate some help. Thanks in advance.

In the single-variable case, $f'(a) \ne 0$ is a sufficient condition for $f$ having an inverse in some neighborhood of $a$. This is not a necessary condition, as it is possible for $f'$ to be $0$ at isolated points and still have local inverses everywhere.