Let $f: U \rightarrow \mathbb{R}$ be continuous on an open set $U \subset \mathbb{R}^{2}$. Suppose that on $U$ the partial derivatives $D_{1} f, D_{2} f, D_{1} D_{2} f$ and $D_{2} D_{1} f$ exist and are continuous. Prove that $D_{1} D_{2} f=D_{2} D_{1} f$ on $U$.

If $f$ is infinitely differentiable, and $m \in \mathbb{N}$, what is the maximum number of distinct $m$-th order partial derivatives that $f$ may have on $U$ ?

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

$$f(x, y)= \begin{cases}\frac{x^{2} y^{2}}{x^{4}+y^{4}} & (x, y) \neq(0,0) \\ 0 & (x, y)=(0,0)\end{cases}$$

Let $g: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be defined by

$$g(x, y)= \begin{cases}\frac{x y\left(x^{4}-y^{4}\right)}{x^{4}+y^{4}} & (x, y) \neq(0,0) \\ 0 & (x, y)=(0,0)\end{cases}$$

For each of $f$ and $g$, determine whether they are (i) differentiable, (ii) infinitely differentiable at the origin. Briefly justify your answers.