We construct functions and stochastic processes for which a functional relation holds between amplitude and local regularity, as measured by the pointwise or local Holder exponent. We consider in particular functions and processes built by extending Weierstrass function, multifractional Brownian motion and the L evy construction of Brownian motion. Such processes have recently proved to be relevant models in various applications. The aim of this work is to provide a theoretical background to these studies and to provide a rst step in the development of a theory for such self-regulating processes.