Given a homeomorphism $f$ on a compact manifold $M$, can you find a $C^1$ diffeomorphism within an arbitrarily small neighborhood of $f$? In other words, can you make an arbitrarily small perturbation to smooth out $f$?
It seems like a natural question, and I would guess the affirmative, but I could not find an answer.
I am working on $S^1$, but would be happy with a more general answer.