Being all sick and such this past couple days, this has me screaming with giggles:
When a system goes from one phase to another, there will generally be a stage where the free energy is non-analytic. This is a phase transition. Due to this non-analyticity, the free energies on either side of the transition are two different functions, so one or more thermodynamic properties will behave very differently after the transition. The property most commonly examined in this context is the heat capacity. During a transition, the heat capacity may become infinite, jump abruptly to a different value, or exhibit a "kink" or discontinuity in its derivative.
Yes. That Kink-alert has me quite curious!