Abuse of notation in GR? $f(x)$ vs. $(f \circ \psi^{-1})(x)$ I see some GR books write $f(x)$ (or even $f(x^\mu)$) when talking about a function on a manifold. But with $f: M \to \mathbb R$ and coordinates defined by a chart $\psi: M \to \mathbb R^n$, shouldn't the notation rather be $(f \circ \psi^{-1})(x)$ when evaluating $f$ at the point corresponding to $x$? The only one I see write it like this is Wald; even Carroll, who uses a chart construction similar to Wald's, writes it the first way a lot.
Is this simply a common abuse of notation, or am I actually missing something?
 A: It's an abuse of notation.  If the chart is given by $(U,x)$ with $x:M\rightarrow \mathbb R^n$ the chart map, then for each $p\in U\subseteq M$ we have
$$f(p) = \bigg(f\circ x^{-1}\bigg)\big(x(p)\big) \equiv f_x\big(x(p)\big)$$
where $f_x :\mathbb R^n \rightarrow \mathbb R$ is the local expression of $f$ in the chart $x$.  When working with functions on manifolds, one almost always works at the chart level with such objects like $f_x$ with the understanding that they "lift" to a well-defined function $f$ at the manifold level. Of course, there are objects appear in charts which do not exhibit this behavior, such as the connection coefficients $\Gamma$. These objects are only defined in a chart, and are not tensorial in nature.
In any case, in my opinion it's pedagogically very important to make the distinction between functions on a manifold and functions in a chart. Once the issue is fully understood, I might be tempted to relax a bit and write fewer symbols, with clarifications added as necessary.
