I have two basic questions about first-order phase transitions:

1) is the susceptibility divergent at a first-order phase transition?

2) if yes, does it diverge in a universal way as in continuous phase transitions?

My understanding is as follows, and I would like to know if it is correct. The susceptibility diverges at a first phase transition, but in a non-universal way. To be concrete, let us consider a transition signaled by spontaneous symmetry breaking, where the system is symmetric above the transition and the symmetry is broken across the transition. If the transition is first order, right at the transition symmetry-broken states will be degenerate with the symmetric state. Then if an infinitesimal field that couples to the order parameter is exerted, a symmetry-broken state will have the lowest energy and the system will be pinned to that state. Therefore, the susceptibility diverges. However, at a first-order phase transition the correlation length does not diverge, so short distance details will be important and the divergence of the susceptibility will depend on them, thus will be non-universal.

I appreciate if anyone tells me whether my understanding is right.