  We study the maximum modulus set, $\mathcal{M}(p)$, of a polynomial $p$. We
are interested in constructing $p$ so that $\mathcal{M}(p)$ has certain
exceptional features. Jassim and London gave a cubic polynomial $p$ such that
$\mathcal{M}(p)$ has one discontinuity, and Tyler found a quintic polynomial
$\tilde{p}$ such that $\mathcal{M}(\tilde{p})$ has one singleton component.
These are the only results of this type, and we strengthen them considerably.
In particular, given a finite sequence $a_1, a_2, \ldots, a_n$ of distinct
positive real numbers, we construct polynomials $p$ and $\tilde{p}$ such that
$\mathcal{M}(p)$ has discontinuities of modulus $a_1, a_2, \ldots, a_n$, and
$\mathcal{M}(\tilde{p})$ has singleton components at the points $a_1, a_2,
\ldots, a_n$.
  Finally we show that these results are strong, in the sense that it is not
possible for a polynomial to have infinitely many discontinuities in its
maximum modulus set.
