The phase transition of the classical two-dimensional Potts model, in particular the order of the transition as the number of components $q$ increases, is studied by constructing renormalization-group transformations on the equivalent one-dimensional quantum problem. It is shown that the block transformation with two sites per cell indicates the existence of a critical ${q}_{c}$ separating the small-$q$ and large-$q$ regions with different critical behaviors. The physically accessible fixed point for $qg{q}_{c}$ is a discontinuity fixed point where the specific-heat exponent $\ensuremath{\alpha}=1$, and therefore the transition is of first order.