  In this paper, we study the value sets of non-permutation polynomial
functions over the residue class ring $\mathbb{Z}/m\mathbb{Z}$. When $m=p^r$ is
a power of some prime $p$, an upper bound is given for the size of the value
set of a polynomial function which is not a permutation. We also show that this
upper bound can be achieved by some integral polynomials. Finally, we
generalize the results for any positive integer $m$ with known prime
decomposition.
