The Courant nodal domain theorem (for Neumann boundary conditions) says that the $n$-th eigenfunction has at most $n$ nodal domains (connected components where the eigenfunction has the same sign. However, Chavel in *Eigenvalues in Riemannian Geometry* seems to point out (P23) that Divergence theorem is used, so the regularity of the nodal set matters, whose proof by Cheng in dimension $\ge 3$ is incomplete. My question is: does it really affect the proof of the nodal domain theorem? More generally, does it affect the application of divergence theorem to a nodal domain (say to functions that are restrictions of $C^{\infty}(\mathbb R^n)$.