Abstract: Given the Hilbert function $u$ of a closed subscheme of a projective
space over an infinite field $K$, let $m_u$ and $M_u$ be,
respectively, the minimum and the maximum among all the
Castelnuovo-Mumford regularities of schemes with Hilbert function~$u$.
I show that, for every integer $m$ such that $m_u \leq m \leq M_u$,
there exists a scheme with Hilbert function $u$ and
Castelnuovo-Mumford regularity $m$. As a consequence, the analogous
algebraic result for an O-sequence $f$ and homogeneous polynomial
ideals over $K$ with Hilbert function $f$ holds too.
Although this result does not need any explicit computation, I also
describe how to compute a scheme with the above requested properties.
Precisely, I give a method to construct a strongly stable ideal
defining such a scheme.