  We consider the problem of numerically integrating functions with hyperplane
discontinuities over the entire Euclidean space in many dimensions. We describe
a simple process through which the Euclidean space is partitioned into
simplices on which the integrand is smooth, generalising the standard practice
of dividing the interval used in one-dimensional problems. Our procedure is
combined with existing adaptive cubature algorithms to significantly reduce the
necessary number of function valuations and memory requirements of the
integrator. The method is embarrassingly parallel and can be trivially scaled
across many cores with virtually no overhead. Our method is particularly
pertinent to the integration of Green's functions, a problem directly related
to the perturbation theory of impurity models. In three spatial dimensions we
observe a speed-up of order $100$ which increases with increasing
dimensionality.
