Equality of mixed partial derivatives (Schwarz theorem)

Define $f: \mathbb {R}^{2} \rightarrow \mathbb {R}$ by

Let (1) $u=x^{2}-2xy-3y^{2}$; (2) $u=x^{y^{2}}$; (3) $u=\arccos \sqrt{\dfrac {x}{y}}$, verify the identity

Let $f(x,y)=xy\dfrac {x^{2}-y^{2}}{x^{2}+y^{2}}$, if $x^{2}+y^{2} \neq 0$ and $f(0,0)=0$. Prove that

Determine whether the second-order partial derivatives of the function exist at the point $(0 ; 0)$

Compute, at the point $(0 ; 0)$, the partial derivatives $f_{x y}^{\prime \prime }, f_{y x}^{\prime \prime }$ of the function

Let the function $f$ and its derivatives $f_{x}^{\prime }, f_{y}^{\prime }, f_{x y}^{\prime \prime }$ be defined in some neighborhood of the point $(x_{0} ; y_{0})$, and let $f_{x y}^{\prime \prime }$ be continuous at this point. Prove that at the point $\left(x_{0} ; y_{0}\right)$ the derivative $f_{y x}^{\prime \prime }$ exists, and

Show that the existence (and even the continuity) of $D_{12} f$ does not imply the existence of $D_{1} f$. For example, let $f(x, y) = g(x)$, where $g$ is nowhere differentiable.

Put $f(0,0) = 0$, and

if $(x, y) \neq (0,0)$. Prove that

Let $E$ be an open set in $\mathbb {R}^{n}$. The classes $\mathscr {C}^{\prime }(E)$ and $\mathscr {C}^{\prime \prime }(E)$ are defined in the text. By induction, $\mathscr {C}^{(k)}(E)$ can be defined as follows, for all positive integers $k$ : To say that $f \in \mathscr {C}^{(k)}(E)$ means that the partial derivatives $D_{1} f, \ldots , D_{n} f$ belong to $\mathscr {C}^{(k-1)}(E)$.

Assume $f \in \mathscr {C}^{(k)}(E)$, and show (by repeated application of Theorem 9.41) that the $k$ th-order derivative

is unchanged if the subscripts $i_{1}, \ldots , i_{k}$ are permuted. For instance, if $n \geq 3$, then

for every $f \in \mathscr {C}^{(4)}$.