  In the authors' previous work, it was shown that if a zero mean curvature
$C^4$-differentiable hypersurface in an arbitrarily given Lorentzian manifold
admits a degenerate light-like point, then the hypersurface contains a
light-like geodesic segment passing through the point. The purpose of this
paper is to point out that the same conclusion holds with just
$C^3$-differentiability of the hypersurfaces.
