This study highlights that alpha-functions used in cubic equations of state must be of class C-2 i.e. that their first (d alpha/dT) and second (d(2)alpha/dT(2)) derivatives must exist and must be continuous, positive (alpha > 0), monotonically decreasing (d alpha/dT < 0), convex (d(2)alpha/dT(2) > 0) and also verify d(3)alpha/dT(3) < 0, for any value of the temperature T. Our proposed "consistency test for alpha-functions" gathers all these conditions. The non respect of one of them can entail low-accuracy prediction of binary phase diagrams involving at least one supercritical compound (this statement is illustrated through the case-studies of the CO2-argon and CO2-decane systems) as well as improper variations of pure-component supercritical properties (h and Cp) with respect to the temperature. Finally, an extensive study of the mostly used alpha-functions described in the open literature is performed and shows that all of them fail this test. Some component-dependent alpha-functions may however pass this test but only if mathematical constraints are added to their parameters. (C) 2016 Elsevier B.V. All rights reserved.