I am interested in what sort of properties are preserved for a function defined on a smooth manifold $M$. Preservation in the sense of a physicist, i.e., invariant under coordinate transformations.
Consider a particular chart $x:U\subset M \to x(U)\subset \mathbb{R}^n$ of an $n$ dimensional smooth manifold $M$. The representation of a function $f:M \to \mathbb{R}$, in the specific chart is $f\circ x^{-1}: x(U)\to \mathbb{R}$.
If I represent the same function in another chart, call it $y$, then I get the following: $$ f\circ x^{-1}=\left(f\circ y^{-1}\right)\circ \left(y\circ x^{-1}\right). $$ And because the chart transition functions $y\circ x^{-1}$ are, for a smooth manifold, smooth, I conclude that if $f$ is smooth in one chart; then it is smooth in any other chart. So it suffices to check smoothness in a coordinate chart. Pretty; and powerful.
I want to know what other properties of functions are preserved in this manner: is injectivity, surjectivity also preserved? If a function acquires the value zero in one chart, does it follow that it acquires the value zero in any other?
This last one doesn't seem to be the case by the above equation, to wit, if $f\circ x^{-1}=0$, say, then I can't seem to claim the same for $f\circ y^{-1}$. This is vastly different to how vector quantities transform under coordinate transformations: they get multiplied by a linear transformation, so if a vector equals zero in one coordinate chart, then it is zero in any other.
Closing, I would like to inquire in what sense a function is invariant under coordinate transformations? Am I really allowed to make such a claim, judging by the fact that even though $f$ is one and the same in the above equation, the representatives are different, $f\circ x^{-1}$ and $f\circ y^{-1}$.?
The statement could be rephrased in the language of sheaf theory. Since the domains of definition of charts of a smooth manifold $ M $ constitute a basis for the topology of $ M $, one can think of $ f\circ x^{-1}: x(U)\to \mathbb{R} $ as the restriction of $ f $ to $ U $. Thus, those properties of functions on $ M $ that define a sheaf are invariant under coordinate transformations and conversely, such as smoothness and continuity.
Obviously, locally injective functions do not form a sheaf. As another example, bounded functions are not a sheaf. While locally constant functions define a sheaf, in particular, zero functions.