Phase Transition, Stability of thermodynamic Potentials

I was studying the stability of thermodynamic potentials. It is obvious that internal energy is a convex function of $(S,V) $, and in case of instable circumenstances, the system adjust in such a way that the global and local convexity law, is well respected. However, It is mentioned that $frac{partial u} {partial s}$ must be a continuous function, thus giving us a new form for the $u(S)$ at constnat $V$ graph. However, my question is: why it has to be that $frac{partial u} {partial s}$ is a continuous function? As for the first order phase transitions we can well have some discontinuities of the first partial derivatives of the potentials.