  We prove that the level sets of a real C^s function of two variables near a
non-degenerate critical point are of class C^[s/2] and apply this to the study
of planar sections of surfaces close to the singular section by the tangent
plane at hyperbolic points or elliptic points, and in particular at umbilic
points.
  We also analyse the cases coming from degenerate critical points,
corresponding to elliptic cusps of Gauss on a surface, where the
differentiability is now reduced to C^[s/4].
  However in all our applications to symmetry sets of families of plane curves,
we assume the C^infty smoothness.
