Let $U \subset \mathbb{R}^{m}$ be a nonempty open set. What does it mean to say that a function $f: U \rightarrow \mathbb{R}^{n}$ is differentiable?

Let $f: U \rightarrow \mathbb{R}$ be a function, where $U \subset \mathbb{R}^{2}$ is open. Show that if the first partial derivatives of $f$ exist and are continuous on $U$, then $f$ is differentiable on $U$.

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

$$f(x, y)= \begin{cases}0 & (x, y)=(0,0) \\ \frac{x^{3}+2 y^{4}}{x^{2}+y^{2}} & (x, y) \neq(0,0)\end{cases}$$

Determine, with proof, where $f$ is differentiable.