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

N derivatives are necessary for order N+1 convergence in quadrature: a converse result