In this study we present a method for approximating the solution of a Singularly Perturbed Boundary Value Problem (SPBVP) containing two parameters (ε 1 ,ε 2 ), which multiply the diffusion coefficient and the convection term, respectively. Moreover, we consider that the convection coefficient and the source term present a discontinuity at an intermediate point. Theoretical bounds for the solution and its derivatives are derived for two complementary cases. A parameter uniform numerical scheme is constructed, which involves an upwind finite difference method with an appropriate piecewise uniform mesh. The error estimation and convergence analysis are presented, which show that the scheme provides a parameter uniform convergence of almost first order. Some numerical examples are discussed to illustrate the performance of the present method. © 2019