Let $f: \mathbb R^n \rightarrow \mathbb R^n$ be a homeomorphism that sends every analytic curve to an analytic curve. Does it send every smooth curve to a smooth curve?
I came to this problem when I was thinking about smoothness classes of homeomorphisms. My experience says that probably the answer should be no (I do not really see a reason to preserve smoothness of curves here), but I have no idea how to construct an example.
