I'm working on an exercise for analysis:
"In the proof of Theorem 9.3.3 [Chain Rule for functions from $\mathbb R^p$ to $\mathbb R^q$], the following fact is used twice: If $A(h)$ is a $q\times p$ matrix whose entries are functions of $h\in\mathbb R^p$ and if $A(h)$ is continuous at $h=0$, then $\lim_{h\to 0}A(h)h=0$, where $A(h)h$ is the result of the matrix $A(h)$ acting via vector-matrix product on the vector $h$. Prove that this limit is $0$, as claimed,"
and it only just occurred to me that the text had been using the idea of continuous matrix-valued functions without defining it, and so now I'm not sure how to go about proving this fact. I assume the definition would be something like each of the columns or entries being continuous, or the resulting matrix-vector product of the matrix function and some specified vector being continuous, but I want to make sure which.