To what extent differentiable mappings of an affine line into a manifold determine its differentiable structure? What about mappings of a plane?

Question

If $M$ is a (real) differentiable manifold, its differentiable structure is completely determined if it is known which mappings $M\to\mathbf{R}$ are differentiable.

How much can be said about the differentiable structure of $M$ if it is known which mappings $\mathbf{R}\to M$ are differentiable? I suspect that this is not enough to determine the differentiable structure of $M$. If so, is there a number $k <\operatorname{dim}M$ such that the differentiable mappings $\mathbf{R}^k\to M$ completely determine the differentiable structure of $M$?

I think my first question can be rephrased as follows: if $f\colon\mathbf{R}^n\to\mathbf{R}$ has the property that for every differentiable $\gamma\colon\mathbf{R}\to\mathbf{R}^n$, the composition $f\circ\gamma\colon\mathbf{R}\to\mathbf{R}$ is differentiable, does this imply that $f$ is differentiable?

I conjecture that if $f\colon\mathbf{R}^n\to\mathbf{R}$ has the property that for every differentiable $\sigma\colon\mathbf{R}^2\to\mathbf{R}^n$, the composition $f\circ\sigma\colon\mathbf{R}^2\to\mathbf{R}$ is differentiable, then $f$ is differentiable.

I do not expect this to be a research level question, but this and a similar questions asked on Mathematics.SE are left without satisfactory answers.

Answer 1

This is true for infinitely differentiable curves: if a map sends smooth curves to smooth curves, then it is smooth, by a theorem of Boman from 1967:

Jan Boman. Differentiability of a Function and of its Compositions with Functions of One Variable. Mathematica Scandinavica 20 (1967), 249–268.

Boman also proves (Theorem 3) that in the case of curves $C^∞$-functions in his theorem cannot be replaced by $C^k$-functions for a finite $k$.

However, if we switch from curves to surfaces, Theorem 8 in the cited paper shows that maps that send $C^k$-surfaces to $C^k$-surfaces are $C^k$-differentiable, for both finite and infinite $k$, answering the original question in the affirmative.