The range of all regularities for polynomial ideals with a given Hilbert function

Let A denote any polynomial ring over a field K and I any homogeneous ideal of A. In this paper it is proven that, given an Hilbert function u, the set of the regularities of the homogeneous ideals I such that the K-algebra A/I has Hilbert function u is an interval of integers. This result is achieved by means of constructive arguments related to the minimal functions with a given Hilbert polynomial and a given regularity.