We formulate the general waveguide discontinuity problem in terms of reflection and transmission operators using a general mode-matching method. This allows the multidiscontinuity problem for arbitrarily shaped waveguides to be solved easily. We then relate the mode-matching method to the integral equation method by using the equivalence principle. This sheds light on the issue of relative convergence. We prescribe a more relaxed condition for obtaining a convergent solution to this problem. The idea is illustrated by a numerical simulation.