  We analyze the finite size scaling of the $q$-state clock model in the $q
\rightarrow \infty$ limit. The behaviors of the specific heat, Binder-Landau
and U4 cumulants agree with the Borgs-Koteck\'y ans\"atz for first order phase
transitions. However, we find that the leading correction to the position of
the extremal points of these quantities is not universal. On the other hand,
the finite size corrections to the mass gap behave like for second order phase
transitions. In particular, the curves corresponding to different size
approximations do not cross in the vicinity of the transition points. The
feature is associated to the existence of a divergent correlation length and
holds for a wider class of models.
