  Copulas are known to satisfy a number of regularity properties, and one might
therefore believe that their densities, when they exist, admit a certain degree
of regularity themselves. We show that this is not true in general by
constructing a broad family of copulas which admit densities that can hardly be
considered regular. The copula densities are constructed from arbitrary
univariate densities supported on the unit interval, and we show by example
that the copula densities can inherit pathological behaviour from the
underlying univariate densities. In particular, we construct a nontrivial
univariate density which is unbounded in every open set of the unit interval,
and show that it induces a copula density which is finite everywhere but
unbounded in every neighborhood of the unit hypercube. Nevertheless, all of our
copulas are shown to enjoy attractive smoothness properties.
