  New proofs of theorems on the multifractal formalism are given. They yield
results even at points q for which Olsen's functions b(q) and B(q) differ.
Indeed, we provide an example of measure for which functions b and B differ and
for which the Hausdorff dimension of the level sets of the local Holder
exponent are given by the Legendre transform of b and their packing dimension
by the Legendre transform of B.
