A long time ago I've asked here about what O'Neill defines in his "Elementary Differential Geometry" book as "Natural Coordinate Functions". In the time, I've understood that it was a notational convenience and all of that. But now I've started to thing again about it, and I just want to make sure I've understood what's happening.
O'Neill in his book, instead of presenting functions as: "let $f: \Bbb R^2 \to \Bbb R$ be given by the relation $f(x,y)=x^2+y^2$" where $x,y \in \Bbb R$ he presents things like "let $f: \Bbb R^2 \to \Bbb R$ be given by the folllowing $f=x^2+y^2$" where he understands $x$ and $y$ to be the functions such that $x(a,b)=a$ and $y(a,b)=b$.
Now, in $\Bbb R^n$ it doesn't really seem necessary to work as that, but now it comes the question: is this the way we usually should setup function in abstract manifolds? Indeed, let $M$ be a smooth manifold of dimension $n$, and let $(x,U)$ be a chart for $M$. We have then $x: U \subset M \to \Bbb R^n$, if $I : \Bbb R^n \to \Bbb R$ is the identity function we can then extend this idea of O'neill's book and define $x^i = I^i \circ x$ to be the $i$-th coordinate function. Now, we can set up every function $f : U \to \Bbb R$ in terms of these functions, so that we can write for instance (if $n = 2$) $f = (x^1)^2 + (x^2)^2$ and so we have a very natural way to write things in $U$ in terms of "the coordinates" $(x^1,\cdots,x^n)$. So that instead of really writing $f(p)$ for $p \in M$ we just write $f$ as a combination of the coordinate functions (using addition, multiplication by scalar, product, composition and so on)?
Is that right? Is really like this that this thing of "coordinate functions" really work?
Thanks very much in advance.