My problem is the following:
Let A be an open subset of $\mathbb{R}^n$ and $C(A)$ be the set of all continuously differentiable functions from $A$ to $ \mathbb{R}^1$. Prove that the set of continuously differentiable functions from $A$ to $\mathbb{R}^m$ (let's call it $C^*$) is a $C(A)$-module.
In order to do so, I know I need to show that for $x\in C(A)$, $y\in C^*$, $xy\in C^*$. I'm assuming this is sort of similar to the product rule and uses the Jacobian but I'm drawing blanks on how to prove it.
$xy$ is in $C^*$ iff each component $xy_i$ is in $C(A)$, and this is true by product rule of scalar functions.