This article is largely concerned with the time-discretization of differential-algebraic equations (DAE) with complementarity constraints, which we name differential algebraic linear complementarity systems (DALCS). Specifically, the Euler implicit discretization of DALCS is analysed: the one-step non-smooth problem (OSNSP), that is a generalized equation, is shown to be wellposed under some conditions, then the convergence of the discretized solutions is studied, and the existence of solutions to the continuous-time system is shown as a consequence. Passivity of some operators is pivotal to the analysis. Examples from circuits, mechanics and switching DAE illustrate the applicability of the developments.