In recent years, there has been a surge of research into methods for estimating derivatives of performance measures from sample paths of stochastic systems. In the case of queueing systems, typical performance measures are mean queue lengths, throughputs, etc., and the derivatives estimated are with respect to system parameters, such as parameters of service and interarrival time distributions. Derivative estimates potentially offer a general means of optimizing performance, and are useful in sensitivity analysis. This paper concerns one approach to derivative estimation, known as infinitesimal perturbation analysis. We first develop a general framework for these types of estimates, then give simple sufficient conditions for them to be unbiased. The key to our results is identifying conditions under which certain finite-horizon performance measures are almost surely continuous functions of the parameter of differentiation throughout an interval. The sufficient conditions we introduce are formulated in the setting of generalized semi-Markov processes, but translate into readily verifiable conditions for queueing systems. These results substantially extend the domain of problems in which infinitesimal perturbation analysis is provably applicable.