In this paper a novel discrete-time implementation of sliding-mode control systems is proposed, which fully exploits the multivaluedness of the dynamics on the sliding surface. It is shown to guarantee a smooth stabilization on the discrete sliding surface in the disturbance-free case, hence avoiding the chattering effects due to the time-discretization. In addition when a disturbance acts on the system, the controller attenuates the disturbance effects on the sliding surface by a factor $h$ (where $h$ is the sampling period). Most importantly this holds even for large $h$. The controller is based on an implicit Euler method and is very easy to implement with projections on the interval $[-1,1]$ (or as the solution of a quadratic program). The zero-order-hold (ZOH) method is also investigated. First and second order perturbed systems (with a disturbance satisfying the matching condition) without and with dynamical disturbance compensation are analyzed, with classical and twisted sliding-mode controllers.