In my textbook it says that for a map $f:\mathbb{R^n} \rightarrow \mathbb{R}^m$, where the partial derivatives are continuous at $a\in \mathbb{R^n}$, we have the the following approximation theorem: $$f(a+h) = f(a) + J_f(a)h + r(h),$$ where $J_f$ is the Jacobian and $\lim_{|h|\rightarrow0}(|r(h)|/|h|)=0$.
Then, right after that, we have a definition that $f$ is differentiable at $a$ if there exists a linear map $L:\mathbb{R^n} \rightarrow \mathbb{R}^m$ such that it holds that $$f(a+h) = f(a) + Lh + r(h), \quad \lim_{|h|\rightarrow0}(|r(h)|/|h|)=0.$$
I don't understand the difference. If we evaluate the Jacobian at a point, we get a linear matrix $J_f(a)$. We have the same conditons for $r$, so then it seems that continuity of the first partial derivatives is sufficient for the map $f$ to be diffeentiable, but as far as I know, this is not the case. Is the first theorem wrong?
