We review the solutions of some functional equations for which the differentiability was the aim of study in the beginning of the last century. By decomposing these functions on the Schauder basis (which are only Lipschitz of order 1), we determine the exact Hölder regularity (even when it exceeds 1) and thus prove that this regularity changes widely from point to point. We also determine the Sobolev (or Besov) spaces to which these functions belong. We finally study the validity of the multifractal formalism which relates some (Sobolev) functional norms of a function to its 'Hölder spectrum' (which is the dimension of the set of points where the function has a given Hölder regularity).