On the nonexistence of $CR$ functions on Levi-flat $CR$ manifolds

We show that no compact Levi-flat $CR$ manifold of $CR$ codimension one admits a Continuous $CR$ function which is nonconstant along leaves of the Levi foliation. We also prove the nonexistence of certain $CR$ functions on a neighborhood of a compact leaf of some Levi-flat $CR$ 3-manifolds, and apply it to showing that some foliated 3-manifolds cannot be embedded as smooth Levi-flat real hypersurfaces in complex surfaces.