Residual Monte Carlo (RMC) methods have been previously used in neutron transport for monoenergetic and multigroup problems. In this paper, we implement an RMC algorithm for solving continuous energy problems with elastic scattering functions. We use a piecewise constant finite-element trial space to approximate the transport solution and build the residual representation. Because of the complexity of the scattering term, an analytic distribution cannot be computed for the residual; instead, we sample source particles directly from the scattering integrand using unnormalized importance sampling. We achieve exponentially convergent Monte Carlo (MC) with the use of an additional weight cancellation technique to reduce the magnitude of particle weights. We then demonstrate the algorithm on continuous energy problems and compare the results with standard MC simulations to demonstrate the increased efficiency of the RMC method.