Is continous symmetry breaking the necessary or adequate condition for Nambu modes ?

Dear all,

I have a question regarding the usual Goldstone theorem, which states that, for a system with continuous symmetry breaking, massless bosons must appear. However, if you look at the derivations of this theorem [1], the crucial assumption seems that, the conserved quantity associated with this symmetry has a local form, i.e., one can define its density and the corresponding current density. As long as this condition is met, the massless modes follow definitely. If so, then the symmetry may not necessarily be continuous, and the conditions can be relaxed as: (1) there exists a symmetry that leaves the Hamiltonian invariant but alters the ground state; (2) the conservable derived from this symmetry has a local form.

May I say that ?

[1]Gene F. Mazenko, Fluctuations, order and defects, p215