  We show that non-occurrence of the Lavrentiev phenomenon does not imply that
the singular set is small. Precisely, given a compact Lebesgue null subset of
the line $E$ and an arbitrary superlinearity, there exists a smooth, strictly
convex Lagrangian with this superlinear growth, such that all minimizers of the
associated variational problem have singular set exactly $E$, but still admit
approximation in energy by smooth functions.
