This paper provides a characterization of the value function arising in optimal control as the unique generalized solution to the Hamilton Jacobi equation (HJE) satisfying a boundary condition. It concerns a class of optimal control problems in which the dynamic constraint is formulated as a differential inclusion and for which the value function is a possibly discontinuous, lower semicontinuous (lsc) function. The key feature of the problems considered is that the velocity set is discontinuous w.r.t. time. The starting point for our work is the HJE analysis of Frankowska and co-authors, who earlier used viability methods to achieve such a characterization of lsc value functions, for measurably time-dependent velocity sets, in terms of "almost everywhere" generalized solutions V to the HJE that possess the following regularity property: the epigraph of V is absolutely continuous w.r.t. time. In this paper we investigate problems for which the velocity set is permitted to be discontinuous w.r.t. time, but when we restrict the nature of discontinuities considered, by requiring the velocity set to be continuous on a set of full measure and having everywhere left and right limits, w.r.t. time. We show that, for this class of problems, it is possible to carry out an HJE analysis based on simpler kinds of generalized solutions and when the "epicontinuity" condition is no longer explicitly imposed. The paper also provides generalizations to allow for state constraints.