In an earlier paper the authors introduced the inverse measure µ^â  (dt) of a given measure µ(dt) on [0,1] and presented the 'inversion formula' f^â  (a) = af(1/a) which was argued to link the respective multifractal spectra of µ and µ^â  . A second paper established the formula under the assumption that µ and µ^â   are continuous measures. Here, we investigate the general case which reveals telling details of interest to the full understanding of multifractals. Subjecting self-similar measures to the operation µ->µ^â   creates a new class of discontinuous multifractals. Calculating explicitly we find that the inversion formula holds only for the 'fine multifractal spectra' and not for the 'coarse' ones. As a consequence, the multifractal formalism fails for this class of measures. A natural explanation is found when drawing parallels to equilibrium measures of dynamical systems. In the context of our work it becomes natural to consider the degenerate Hölder exponents 0 and infinity.