An asymptotic holomorphic boundary problem on arbitrary open sets in Riemann surfaces

We show that if U is an arbitrary open subset of a Riemann surface and φ an arbitrary continuous function on the boundary ∂U, then there exists a holomorphic function φ˜ on U such that, for every p∈∂U, φ˜(x)→φ(p), as x→p outside a set of density 0 at p relative to U. These “solutions to a boundary problem” are not unique. In fact they can be required to have interpolating properties and also to assume all complex values near every boundary point. Our result is new even for the unit disc.