State the inverse function theorem for maps $f: U \rightarrow \mathbb{R}^{2}$, where $U$ is a non-empty open subset of $\mathbb{R}^{2}$.

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be the function defined by

$$f(x, y)=\left(x, x^{3}+y^{3}-3 x y\right) .$$

Find a non-empty open subset $U$ of $\mathbb{R}^{2}$ such that $f$ is locally invertible on $U$, and compute the derivative of the local inverse.

Let $C$ be the set of all points $(x, y)$ in $\mathbb{R}^{2}$ satisfying

$$x^{3}+y^{3}-3 x y=0$$

Prove that $f$ is locally invertible at all points of $C$ except $(0,0)$ and $\left(2^{2 / 3}, 2^{1 / 3}\right)$. Deduce that, for each point $(a, b)$ in $C$ except $(0,0)$ and $\left(2^{2 / 3}, 2^{1 / 3}\right)$, there exist open intervals $I, J$ containing $a, b$, respectively, such that for each $x$ in $I$, there is a unique point $y$ in $J$ with $(x, y)$ in $C$.