Families of variational functionals exist admitting the use of discontinuous trial functions in the approximate solution of neutron diffusion and transport equations. In this paper the fundamental properties of these approximation procedures are discussed. The methods of diffusion theory considered cover synthesis and finite dimensional approximations, continuous and multigroup energy dependence, and independent approximations of flux and current as well as approximations of the diffusion equation by using Lagrange multipliers. Assuming the Galerkin weighting, conditions assuring the stability and convergence of these methods are given. Reasoning by analogy, similar results are obtained for methods of transport theory.