Abstract: We consider domains whose boundary can be locally represented as the graph of a continuous function and construct smooth approximations that preserve topological properties (in particular the fundamental group, for instance). The main tool for doing this is a notion of (multivalued) map of "good directions at a point", that is a map that associates to a point in the neighbourhood of the boundary the directions along which the boundary can be locally represented as the graph of a continuous function. We study various properties of the map of good directions and also use it to show that there must be points on the the boundary of the domain, in a neighbourhood of which the domain is in fact smoother, it is locally Lipschitz. This is joint work with John M. Ball.