In a book by Robert Hermann, chapter 4, the author tries to specialize the general theory of differentiable manifolds to plain $\mathbb{R}^n,$ and he distinguishes between $n-$tuples $(x_j),$ $1\leq j \leq n,$ which he calls points and $x_j$ without parentheses, which are the real-valued functions assigning to $(x_j)$ the $j$th coordinate of $(x_j).$ Then he makes the remark that the reader should note that much of the notational confusion in undergraduate differential calculus is caused by not making this distinction. My questions are:
1) If the points $(x_j)$ are $n-$tuples, is it meant that the $j$th coordinate that the author is referring to in the definition of $x_j$ above is nothing more than the $j$th slot in the ordered $n-$tuple? Or is he in fact presupposing a vector space structure on $\mathbb{R}^n$ and the coordinate $x_j$ is really a coordinate of a vector $(x_j)$ with respect to (canonical) basis? And is this distinction important at all?
2) What are some good examples illustrating the point that not making the distinction between coordinates and coordinate functions can lead to confusion?