  Results on the error bounds of quadrature methods are well known - most state
that if the method has degree N, and the integrand has N derivatives, then the
error is order N+1. We prove here a converse: that if the integrand fails to
have N derivatives, even only at a finite number of points, no method,
regardless of its degree, can guarantee convergence more than order N. Even if
the integrand fails to have N derivatives at just 3 (for even N, 2) points, no
method can produce order more than N+1 convergence. This is done by an
adversarial proof: we explicitly construct the functions that exhibit such
error; simple splines turn out to suffice.
