We have used the Monte Carlo renormalization group method to study the order‐disorder phase transition of the three and four dimensional, three‐state Potts model. In both cases, the transition appears to be first order. However, no evidence for a discontinuity fixed point associated with these transitions was found. Instead, the existence of a nearby second order fixed point just beyond the first order transition could be inferred from the RG flow in the ordered as well as in the disordered state. These results imply that the thermodynamic quantities are analytically continuable into the metastable regions, and that apart from discontinuities caused by the intersection of the two free‐energy branches, no additional singularities occur at the transition.