Suppose I have a $C^{\infty}$ map $f:X\to\mathbb{R}^{n}$, for some differential manifold $X$.
Then I also have a homeomorphic coordinate map $h$ from a subset of $\mathbb{R}^{n}$ to a subset of $X$.
My question is, since $f$ is $C^{\infty}$, then are necessarily $f\circ h$ going to be $C^{\infty}$?
I would think the answer is yes since homeomorphisms preserve convergence, but I'm hesitant to conclude so because I don't know if homeomorphisms are necessarily differentiable.