Uniform approximation of homeomorphisms by diffeomorphisms

We prove that a compactly supported homeomorphism of a smooth manifold of dimension n≥5n≥5 can be approximated uniformly by compactly supported diffeomorphisms if and only if it is isotopic to a diffeomorphism. If the given homeomorphism is in addition volume preserving, then it can also be approximated uniformly by volume preserving diffeomorphisms.