(introduction)
First of all, it is a bit tricky to start this question by defining the terms, because the idea of "smooth structure" is, I think, a bit too spread out to contain its essence in a single definition. I see it more like a set of equivalent definitions, where each definition portrays an equally important property of it. And this is the whole point of my problem here.
(the root of the problem)
A usual way to introduce a smooth structure on $M$ is by an atlas (where we can assume that $M$ is a topological manifold or just a set, depending on the style). Then, for the sake of uniqueness we can require the atlas to be maximal (so now we can call it the smooth structure), but this maximality is far from a practically useful construct...
(the part that I do understand)
...That's why I searched more and realized that the set of all smooth functions $M\rightarrow\mathbb{R}$ distinguish the smooth structure (two different smooth structures on $M$ have different sets of smooth functions). That was actually useful for some lemmas. This is even easy to prove: if two atlases don't match, then, by the definition of it, the transition map between some two of them is not smooth and so at least one coordinate of it is not smooth and that's the function which is smooth for one but not for the other smooth structure.
(actual question)
But then, for some other lemmas even this interpretation of the smooth strucutre was of no help at all, but I hypothesised another. That is: the set of all smooth functions $\mathbb{R}\rightarrow M$ distinguish the smooth structure too (assuming the topology is already given). It would be of help in proving those other statements, but I can't seem to progress in analyzing the idea. Both proving it (by construction of a diffeomorphism or otherwise) and constructing a counterexample are difficult. Also, searching on the internet was not fruitful so I ended up here to ask: is it true and, if so, is it a known fact?
