Differentiability criterion

Differentiability criterion: Let $U\subseteq\mathbb R^k$ be an open set, let $f:U\to\mathbb R^m$ , and fix $a\in U$ . The following are equivalent:

$f$ is differentiable at $a$ (in the Fréchet sense).
There exists a linear map $A:\mathbb R^k\to\mathbb R^m$ such that $\lim_{h\to 0}\frac{\|f(a+h)-f(a)-Ah\|}{\|h\|}=0.$

In this case, $A$ is unique and is denoted $Df(a)$ .

Remarks

This formulation says that differentiability is exactly the existence of a first-order linear approximation with an error term that is $o(\|h\|)$ . In the one-dimensional case $k=m=1$ , it is equivalent to the existence of the usual derivative at $a$ .