Please help to solve part C with proving that dG is continuous

" 7.7 The entropy of a certain fluid has been found to be re- lated to its internal energy and volume in the following way: U V S = S + a In + 8 In U V where S, U, and V are, respectively, the molar en- tropy, internal energy, and volume of the fluid for some appropriately chosen reference state, and a and are positive constants. a. Develop an interrelationship between internal en- ergy, temperature, and specific volume (the thermal equation of state) for this fluid. b. Develop an interrelationship between pressure, tem- perature, and volume (the volumetric equation of state) for this fluid. c. Show that this fluid does not have a first-order phase transition by establishing that the fluid is stable in all thermodynamic states