The one-shot shortest path query has been studied for decades. However, in the applications on road networks, users are actually interested in the path with the minimum travel time (the fastest path), which varies as time goes. This motivates us to study the continuous evaluation of fastest path queries in order to capture the dynamics of road networks. Repeatedly evaluating a large number of fastest path queries at every moment is infeasible due to its computationally expensive cost. We propose a novel approach that employs the concept of the affecting area and the tolerance parameter to avoid the reevaluation while the travel time of the current answer is close enough to that of the fastest path. Furthermore, a grid-based index is designed to achieve the efficient processing of multiple queries. Experiments on real datasets show significant reduction on the total amount of reevaluation and therefore the cost for reevaluating a query.