'Chemical' First-order Phase Change
Well, let's see. As you know, the equilibrium constant is determined by the reaction's $\Delta_rG:\;K=e^{-\frac{\Delta G}{RT}}$, so a discrete jump in the former means a discrete jump in the latter. What is that $\Delta_rG$? That's the sum of $\Delta_fG$ of products minus that of starting compounds. Now what is the $\Delta G$ of formation of a compound? A continuous function, unless there is a phase transition, which by your problem definition is not the case. Well, a linear combination of a few continuous functions is still a continuous function.