Let $$f(x,y) = \begin{cases} \frac{x^3 y^3}{x^2 + y^2}, & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases}$$ Choose the correct statement(s) from below:
(A) $f$ is continuous on $\mathbb{R}^2$;
(B) $f$ is continuous at every point of $\mathbb{R}^2 \backslash \{(0,0)\}$;
(C) $f$ is differentiable at every point of $\mathbb{R}^2 \backslash \{(0,0)\}$;
(D) $f$ is not differentiable at $(0,0)$.

Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. If $i$, $j$ and $k$ are three integers in $\llbracket 1, n \rrbracket$, the second partial derivative of $f_k$ at $x$ with respect to the variables $x_i$ and $x_j$ is denoted $f_{i,j,k}(x)$.
We assume that the Jacobian matrix $J_f(x)$ is antisymmetric for all $x$ in $\mathbb{R}^n$.
Show that there exist a real square matrix $A$ of size $n$ and an element $b$ of $\mathbb{R}^n$ such that for all $x$ in $\mathbb{R}^n$, $f(x) = Ax + b$. Justify that $A$ is antisymmetric.

Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. What is the necessary and sufficient condition on $f$ for the Jacobian matrix $J_f(x)$ to be antisymmetric for all $x$ in $\mathbb{R}^n$?

Now $f$ is a function of class $C^1$ from $\mathbb{R}^n$ to itself.
Show that the Jacobian matrix $J_f(x)$ is symmetric for all $x$ in $\mathbb{R}^n$ if and only if there exists $g$ of class $C^2$ on $\mathbb{R}^n$ with values in $\mathbb{R}$ such that $$\forall x \in \mathbb{R}^n, \forall i \in \llbracket 1, n \rrbracket, \quad f_i(x) = \mathrm{D}_i g(x)$$
One may consider the map $g$ defined by $g(x) = \sum_{i=1}^n x_i \int_0^1 f_i(tx)\, \mathrm{d}t$ and express $\mathrm{D}_i g(x)$ as a single integral.