# Proof that equilibrium phase fraction(s) is a continuous function of composition

It seems that the equilibrium phase fractions (i.e., relative amounts of phase A1, A2, ..., Aq for a system in which there are q total phases) as a function of composition (for a constant pressure and temperature) is a continuous function. I haven't been able to "prove" it rigorously. I'm hoping that someone can direct me to a "proof" of this somewhere in the literature.

• Phase fraction isn't even a function of pressure and temperature, even for a pure substance. If I hold 1kg of pure water at the (usual) triple point, what fraction of the water is solid, what fraction is liquid, and what fraction is gas? (Unknowable without further state information.) Commented May 21 at 4:42
• If I hold 1kg of pure iron in the Earth's magnetic field, what fraction of the iron is in magnetised phase, and what fraction isn't? (Unknowable, and path-dependent at that.) Commented May 21 at 4:45
• @ShernRenTee yes so this gets at something interesting. At very select combinations of temperature and pressure (those corresponding to coexistence line/points on a P-T diagram) the phase fractions are not well defined. But when one restricts things to constant P-T combinations where phase fraction is well-defined (which is in some sense "nearly every" case) then it seems to be a constant function of composition. Commented May 21 at 18:00
• The proof that "wherever the phase fraction is a function of composition and doesn't have any discontinuities, it is a continuous function of composition" is left as an exercise for the reader. (: Commented May 21 at 21:21
• Then you are on track to conjecture: every "discontinuity" in equilibrium composition, as a function of continuous change in other parameters, is actually a phase transition, so that wherever the composition would be discontinuous it is in fact not even defined. This seems like one of those multivariable calculus exercises where nine tenths of the work is in "setting out", and once you have somehow mathematized the problem the answer falls into your lap. Commented May 23 at 3:49