I have this claim left as an exercise in my course:
Let $f:M\to N$ be some function between two smooth manifolds $M$ and $N$ (respectively of dimensions $m$ and $n$). Prove that, if for any smooth function $\mu:N\to\mathbb{R}$, we have that $\mu\circ f$ is smooth, then $f$ is smooth. (Here, I say smooth to say "smooth of class $C^{k}$", $k\geq 1$). Hint: use the chain rule.
Here is how I tried:
Let $(U_{i},\varphi_{i})_{i\in I}$ be some atlas of $M$ and $(V_{j},\psi_{j})_{j\in J}$ some atlas of $N$. Now take $(i,j)\in I\times J$ such that $f^{-1}(V_{j})\cap U_{j}\neq\emptyset$. Proving that $f$ is smooth means that we need to show \begin{equation}\psi_{j}\circ f\circ\varphi^{-1}_{i}\left.\right\vert_{\varphi_{i}(f^{-1}(V_{j})\cap U_{i})}\end{equation} is of class $C^{k}$ for any $(i,j)\in I\times J$.
We know that every component of $\psi_{j}$ is smooth and then, by hypothesis on $f$, any component of $\psi_{j}\circ f$ is smooth. I posted that on another forum and their conclusion was I can't state that without constructing a function extending $\psi_{j}$ $C^{k}$-continuously on $N$. We can do that by constructing a function $$\begin{aligned}\alpha_{j}&=1 &\text{on}\,\,V'_{j}\\ &= 0 &\text{out of}\,\,V_{j} \end{aligned}$$ and $\alpha_{i}$ is some $C^{k}$-continuous function on $V_{j}\setminus V'_{j}$ where $V'_{j}$ is some open set of $N$ such that $\overline{V'}_{j}\subset V_{j}$. If this is true until there, my main problem is to prove the existence of such an open set $V'_{j}$.
I suppose the chain rule is there to prove the smoothness of the composition in the first equation above.