Past exam of the mathematics course of the University of Cambridge / 2004 / ib / Paper 4 / 3F (Analysis II)

The map is differentiable at $u$ when there exists a linear map $A:{\mathbb{R}}^n\to {\mathbb{R}}^m$ such thatThe 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 $ab\ne 0$, choose $\delta <\min (|a|,|b|)$. If $\Vert (h,k)\Vert <\delta$, neither coordinate changes sign, andThus the remainder in the differentiability definition is identically zero throughout that neighborhood. Consequentlywhich proves the requested differentiability directly.