State precisely the inverse function theorem for a smooth map $F$ from an open subset of $\mathbf{R}^{2}$ to $\mathbf{R}^{2}$

Define $F: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}$ by

$$F(x, y)=\left(x^{3}-x-y^{2}, y\right)$$

Determine the open subset of $\mathbf{R}^{2}$ on which $F$ is locally invertible.

Let $C$ be the curve $\left\{(x, y) \in \mathbf{R}^{2}: x^{3}-x-y^{2}=0\right\}$. Show that $C$ is the union of the two subsets $C_{1}=\{(x, y) \in C: x \in[-1,0]\}$ and $C_{2}=\{(x, y) \in C: x \geqslant 1\}$. Show that for each $y \in \mathbf{R}$ there is a unique $x=p(y)$ such that $(x, y) \in C_{2}$. Show that $F$ is locally invertible at all points of $C_{2}$, and deduce that $p(y)$ is a smooth function of $y$.

[A function is said to be smooth when it is infinitely differentiable.]