  Let $A = K[x_1, ..., x_n]$ denote the polynomial ring in $n$ variables over a
field $K$ with each $\deg x_i = 1$. Let $I$ be a homogeneous ideal of $A$ with
$I \ne A$ and $H_{A/I}$ the Hilbert function of the quotient algebra $A / I$.
Given a numerical function $H : \mathbb{N} \to \mathbb{N}$ satisfying
$H=H_{A/I}$ for some homogeneous ideal $I$ of $A$, we write $\mathcal{A}_H$ for
the set of those integers $0 \leq r \leq n$ such that there exists a
homogeneous ideal $I$ of $A$ with $H_{A/I} = H$ and with $\depth A / I = r$. It
will be proved that one has either $\mathcal{A}_H = \{0, 1, ..., b \}$ for some
$0 \leq b \leq n$ or $|\mathcal{A}_H| = 1$.
