Pointwise smoothness of space-filling functions

We study irregularity properties of generic Peano functions; we apply these results to the determination of the pointwise smoothness of a Peano function introduced by Lebesgue and of some related functions, showing that they are either monohölder or multifractal functions. We test on these examples several numerical variants of the multifractal formalism, and we show how a change of topology on R can affect the Hölder regularity of such functions.