This paper concerns the characterization of the value function associated with a state constrained Bolza problem in which the data are allowed to be discontinuous w.r.t. the time variable on a set of zero measure and have everywhere left and right limits. Using techniques coming from viability theory and nonsmooth analysis, we provide a characterization of the value function as the unique solution to the Hamilton-Jacobi equation, in a generalized sense which employs the lower Dini derivative and the proximal normal vectors.