Local variability of non-smooth functions

With the help of Müntz powers a formula of the Taylor type for non-smooth functions is presented. The approximation provides a local study for the variability of some curves which do not have a derivative. The approach includes the classical case but, at the same time, other non-analytical and non-differentiable mappings. In the first place, a Müntz curve representing the local variability of a function is defined. The coefficient and exponent of the model allow a numerical characterization of the relative extremes and differentiability of the map. The introduction of exponents of higher order provides a generalization of the Taylor’s formula including some cases of non-differentiability. In the last part, a series expansion of non necessarily integer powers representing the function is presented. Several properties of convergence, continuity, integrability and density are studied.