Efficient computational methods for system reliability assessment are of importance in many contexts, where crude Monte Carlo simulation is inefficient or infeasible. These methods include a variety of importance sampling techniques as well as subset simulation. Most of these methods function in an adaptive manner, whereby the sampling density gradually approaches the failure domain. The adaptation can work well when the limit state function describing system performance is continuous. However, many system reliability problems involve limit state functions that are non-continuous over the input sample space. Such situations occur in both connectivity- and flow-based problems, due to the binary or multi-state random variables entering the definition of the system performance or the discontinuous nature of the performance function. When solving this kind of problem, the standard subset simulation algorithm with fixed intermediate conditional probability and fixed number of samples per level can lead to significant errors, since the discontinuity of the output can result in an ambiguous definition of the sought percentile of the samples and, hence, of the intermediate domains. In this paper, we propose an adaptive subset simulation algorithm to determine the reliability of systems with discontinuous limit state functions. The proposed algorithm chooses the number of samples and the conditional probability adaptively. Numerical examples are provided to demonstrate the accuracy and efficiency of the proposed algorithm.