Definition of $C^1$ functions with values in $\mathbb R^m$

Question

My analysis textbook defines a $C^1$ function $f:\mathbb{R^n}\to\mathbb{R^m}$ as one in which for each component function $f_i, 1\leq i\leq m$ the partial derivative $\frac{\partial f_i}{x_j}$ exists and is continuous for $1\leq j\leq n$. This implies that $f$ is differentiable and the derivative is equal to the Jacobian matrix (which has continuous functions $\frac{\partial f_i}{x_j}$ as its entries). My questions are $(1)$ how would you formulate the notion of continuity for a matrix function $f$? I know that a function from one topological space to a product space is continuous iff each component function is continuous, but I don't know how to experess this in terms of matrices. $(2)$ Assuming $f:\mathbb{R^n}\to\mathbb{R^m}$ is differentiable, and the matrix $Df$ is continuous, would that imply that each of its entries $\frac{\partial f_i}{x_j}$ is are continuous? So I'm trying to see an equivalence in the definition of $C^1$ given in my book, and by examining $Df$ itself.

Answer 1

For the purpose of giving topology to the space of $m\times n$ matrices, one "flattens" the matrix into a vector by stacking its columns. When viewed this way, the space of matrices is just $\mathbb R^{mn}$ with another notation. So the topology is that of $\mathbb R^{mn}$, which is the product topology. As you know, the continuity of a map into a product space can be studied component-wise.

So the conclusion is: yes, the continuity of $Df$ as a map into matrix space is equivalent to the continuity of each entry of $Df$. And these entries are precisely the first-order partial derivatives.