Elitzur's theorem, stating that spontaneous breakdown of a gauge symmetry is impossible, was originally proved for a lattice gauge theory. Is it valid in continuum field theory? Any ref?
Well, you have to specify what you mean by continuum gauge theory. The only way I know how to regulate gauge theories in the continuum directly is the perturbative way, which needs gauge fixing, which already breaks gauge symmetry. In this kind of contexts what can break is a global symmetry, not a local one (this is the Higgs mechanism, which is often sloppily called spontaneous gauge symmetry breaking).
I remember thinking about this question once. :) It particularly bothered me, because you would need symmetry breaking in an SU(2)-Higgs model in particle physics. And since you can define nonperturbatively such QFT by taking a continuum limit of the lattice gauge theory, you seem to have a problem, you seem to always have no symmetry breaking at all. It is easy to show by formal manipulations of the path integral that the Higgs VEV is always zero.
The only solution I could think of in this context was the same as for spontaneous symmetry breaking in general. I think the only way you can see symmetry breaking is to include an explicit breaking term, than do the infinite volume and continuum limit first, and than do the symmetry breaking term goes to zero limit. It is important that the order of the limits is not interchangeable. If you do it the other way you always get zero. Unfortunately I have not seen this kind of calculation carried out anywhere, but this is my best guess.
If anyone has a better understanding of this, I am also very interested.