Abstract: Feedback systems with discontinuous nonlinearities are still far to be fully analyzed and understood. Practical applications show that the behavior of such systems can be “smoothed” by injecting at the input of the nonsmooth nonlinearity a high frequency signal (the dither), e.g. stick-slip compensation in mechanical systems with friction, pulse width modulation for power converters, quenching limit cycles in relay feedback systems, chattering mitigation in sliding mode controlled systems. Averaging theory can provide valuable analytical tools for a rigorous analysis of such phenomenon. For the case when the original nonlinearity is Lipschitz continuous, the averaging of dithered systems was rigorously justified by Zames and Shneydor and Mossaheb. In this seminar it is shown that also discontinuous nonlinearities of feedback systems can be narrowed using dither, as long as the amplitude distribution function of the dither is absolutely continuous and has bounded derivative. The averaged system is proven to approximate the dithered system with an error of order of the dither period. The averaging result is also exploited for the design of a chattering reduction technique in sliding mode controlled systems, when the actuator has a natural on-off behavior. The injected dither signal allows to regulate the boundary layer by ensuring a constant switching frequency of the control signal.