We study the boundary behaviour of holomorphic functions in the Hardy-Sobolev spaces ${\mathscr{H}}^{p,k}\left(\right)$, where is a smooth, bounded convex domain of finite type in ℂⁿ, by describing the approach regions for such functions. In particular, we extend a phenomenon first discovered by Nagel-Rudin and Shapiro in the case of the unit disk, and later extended by Sueiro to the case of strongly pseudoconvex domains.