Let $U, V$ be open sets in $\mathbb{R}^{n}, \mathbb{R}^{m}$, respectively, and let $f: U \rightarrow V$ be a map. What does it mean for $f$ to be differentiable at a point $u$ of $U$ ?

Let $g: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be the map given by

$$g(x, y)=|x|+|y|$$

Prove that $g$ is differentiable at all points $(a, b)$ with $a b \neq 0$.