Let $U \subseteq {\rm I\!R}^{n}$ be an open subset and let $f: U \longrightarrow {\rm I\!R}^{m}$ be a function, where $m \geqslant 1$. Assuming the Jacobian matrix $\textbf{J}$ of $f$ exists, when is it continuous and how would one best show continuity? I know the Jacobian can be thought of as the following mapping:
$\textbf{J}: U \longrightarrow Hom({\rm I\!R}^{n}, {\rm I\!R}^{m}), \ x_{0} \mapsto(v \mapsto \textbf{J}_{x_{0}}(v))$
and $\textbf{J}_{x_{0}}: {\rm I\!R}^{n} \longrightarrow{\rm I\!R}^{m}$ is continuous, since it is a linear map on a finite-dimensional vector space. But when is $\textbf{J}$ continuous in $U$ and how does one show this? I already know that if all partial derivatives exist and are continuous, then $f$ is continuously differentiable, i.e. $\textbf{J}$ is continuous. So does that mean one could check continuity by verfying that $\partial_{j}f_{k}$ is continuous for all $j \in \{1,...,n \}$ and $k \in \{1,...,m \}$, i.e. if every component of $\textbf{J}$ is continuous, then $\textbf{J}$ is continuous? Surely not?
EDIT: Given that vector-valued functions are continuous if and only if their component functions are continuous and the partial derivative $\partial_{k} f: U \longrightarrow {\rm I\!R}^{m}$, $x \mapsto \partial_{k} f(x) = (\partial_{k} f_{1}(x), ..., \partial_{k} f_{m}(x))$ is a vector-valued function for all $k \in \{1,...n \}$, perhaps it isn't such a bad idea after all to deduce continuity of the Jacobian by checking every component, since the columns of the Jacobian are exactly the vector-valued partial derivatives. Any objections?
This is correct.
Perhaps you were just slightly unconvinced that checking continuity of a "simple" object like $\partial_jf_k: U \to \Bbb{R}$ (for all $j,k$) implies the continuity of a more "complicated" object like $\boldsymbol{J}: U \to \text{Hom}(\Bbb{R}^n, \Bbb{R}^m)$.
In fact, using induction, you can replace "continuously differentable" with $\mathcal{C}^k$, for any $k \in \Bbb{N}$.