Let $F$ be a coherent sheaf on a scheme $X$. If $X$ is proper, then there exists a smallest integer $R$ such that $\mathrm{rank} \ F(x)\leqslant R$ for all $x$ in $X$. On the other hand there is also a maximal integer $r$ such that $r\leqslant\mathrm{rank} \ F(x)$ for all $x$ in $X$.
I call $F$ of type $(\ast)$ if for every integer $n$ in $[r,R]$ there exists a point $x$ such that $F(x)$ has rank $n$. It is clear that locally free sheaves satisfy $(\ast)$; on the other hand there are sheaves which do not satisfy $(\ast)$, e.g. take $F=\mathcal{O}_p^{\oplus 2}$, where $p$ is a point.
Is there a description of the class of sheaves satisfying $(\ast)$? What are some nice (non-locally free) sheaves satisfying $(\ast)$. E.g. does the cotangent sheaf $\Omega_X$ satisfy $(\ast)$?