Existing adaptive inverse compensation methods for cancelling actuator backlash nonlinearity are all restricted to handle constant backlash parameters. In other words, when discontinuity and time variation as both ubiquitous phenomena in practical actuators exist, such inverse compensation methods are no longer applicable theoretically. So far, no result has been reported in addressing such an issue, regardless of its importance in practice. In this paper, we solve this problem by developing a new piecewise Lyapunov function analysis and using parameter projection adaptation mechanism. Based on such approaches, an adaptive inverse compensation control scheme is designed to compensate for piecewise time-varying actuator backlash nonlinearity. It is proved that all signals of closed-loop system are ensured bounded. Moreover, the steady-state error is bounded by an adjustable scalar approaching to zero arbitrarily. Simulation also illustrates the obtained theoretical results.