We consider the single-leg airline revenue management problem in continuous time with Poisson arrivals. Earlier work on this problem generally uses the Hamilton–Jacobi–Bellman equation to find an optimal policy whenever the value function is differentiable and is a solution to this equation. In this paper, we employ a different probabilistic approach, which does not rely on the smoothness of the value function. Instead, we use a continuous-time discrete-event dynamic programming operator to construct the value function and study its properties. A by-product of this approach is the analysis of the differentiability of the value function. We show that differentiability may break down for example with discontinuous arrival intensities. Therefore, one should exercise caution in using arguments based on the differentiability of the value function and the Hamilton–Jacobi–Bellman equation in general.