Q1: Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ be the function given by $f(x_1, x_2) = 2 x_1 x_2^7$. Is Schwarz’s theorem applicable?

A1: Yes, because all partial derivatives of arbitrary order exist.

A2: No, because the second-order partial derivatives are not continuous.

A3: One needs more information.

Q2: Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ be the function given by $f(x_1, x_2) = 2 x_1 x_2$. What is the partial derivative $\frac{\partial^2 f}{ \partial x_2 \partial x_1}(x)$

A1: $2$

A2: $1$

A3: $0$

A4: $3$

Q3: What is the correct formulation for Schwarz’s theorem for $f: U \rightarrow \mathbb{R}$ with open set $U \subseteq \mathbb{R}^n$?

A1: If all second-order partial derivatives exist and they form continuous functions $U \rightarrow \mathbb{R}$, then $$ \frac{\partial^2 f}{ \partial x_j \partial x_i}(x) = \frac{\partial^2 f}{ \partial x_i \partial x_j}(x)$$ for all $i,j$ and $x \in U$.

A2: If all second-order partial derivatives exist, then $$ \frac{\partial^2 f}{ \partial x_j \partial x_i}(x) = \frac{\partial^2 f}{ \partial x_i \partial x_j}(x)$$ for all $i,j$ and $x \in U$.

A3: If all second-order partial derivatives exist at one point, then $$ \frac{\partial^2 f}{ \partial x_j \partial x_i}(x) = \frac{\partial^2 f}{ \partial x_i \partial x_j}(x)$$ for all $i,j$ and $x \in U$.

A4: If all second-order partial derivatives exist at one point, then $$ \frac{\partial^2 f}{ \partial x_j \partial x_i}(x) = \frac{\partial^2 f}{ \partial x_i \partial x_j}(x)$$ for the given point $x \in U$.