Differentiable map (ℝ^k→ℝ^m)

Let $U\subseteq\mathbb{R}^k$ be open and let $f:U\to\mathbb{R}^m$. The map $f$ is differentiable on $U$ if it is differentiable at every point $a\in U$ in the Fréchet sense , i.e. if for each $a$ there exists a linear map $Df(a):\mathbb{R}^k\to\mathbb{R}^m$ such that

Differentiability in this sense is the correct multivariable analogue of one-variable differentiability . It implies continuity and yields the multivariable chain rule.

Examples:

• Any polynomial map $f:\mathbb{R}^k\to\mathbb{R}^m$ is differentiable on $\mathbb{R}^k$.
• If $f$ is $C^1$ on $U$ (continuous first partial derivatives), then $f$ is differentiable on $U$.
• The map $f:\mathbb{R}^2\to\mathbb{R}$, $f(x,y)=|x|$ is differentiable at points with $x\ne 0$, but not differentiable along the line $x=0$.