Let $U \subset \mathbb{R}^{m}$ be an open set and $f: U \longrightarrow \mathbb{R}^{n}$. Then, $f$ is differentiable in $a \in U$ if only if for all $h \in \mathbb{R}^{m}$ with $a + h \in U$, theres exists a linear transformation $A(h): \mathbb{R}^{m} \longrightarrow \mathbb{R}^{n}$ such that $f(a+h) - f(a) = A(h)h$ and $h \longmapsto A(h)$ be continuous in $h=0$.
For "$\Longrightarrow$", I know that $$f(a+h)-f(a) = \mathrm{D}f(a)h + \frac{r(h)}{|h|}$$ where $\displaystyle \lim_{h\to 0}\frac{r(h)}{|h|} = 0$. But I cannot define the transformation $A(h)$. Someone can helps me? I like hints, no solutions.
Take $A(h)=Df(a)$ for every $h$. For the converse part show that partial derivatives exist and are continuous at $a$.