Mathematical properties of free Gibbs energy
n is discontinuous. The least amount you can change it by is ~1 /6.022E23 of 1 mole. You'll have to be more specific about what type of (mathematical) continuity you are referring to, but it doesn't seem useful to ask about its fine grained continuity since it is a macroscopic concept.  Statistical (quantum) Thermodynamics is used both for microscopic systems and as the basis for macroscopic quantities such as Gibbs Free Energy. You need to be careful: GFE is one meaning of Thermodynamic Potential, but not the only one. I don't have any idea what you mean by G's coercivity.