= Solution
The map is <differentiable> at $u$ when there exists a <linear map> $A:\mathbb R^n\to\mathbb R^m$ such that
$$
\boxed{\lim_{h\to0}\frac{\|f(u+h)-f(u)-Ah\|}{\|h\|}=0.}
$$
The increments are taken sufficiently small to stay in the open domain; $A=Df(u)$ is the derivative. This is a joint approximation in all directions, not just the existence of coordinate partial derivatives.
At $(a,b)$ with $ab\ne0$, choose $\delta<\min(|a|,|b|)$. If $\|(h,k)\|<\delta$, neither coordinate changes sign, and
$$
g(a+h,b+k)-g(a,b)=\operatorname{sgn}(a)h+\operatorname{sgn}(b)k.
$$
Thus the remainder in the differentiability definition is identically zero throughout that neighborhood. Consequently
$$
\boxed{Dg(a,b)(h,k)=\operatorname{sgn}(a)h+\operatorname{sgn}(b)k,}
$$
which proves the requested differentiability directly.
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