We consider a class of state constrained Bolza problems in which the integral cost is merely continuous w.r.t. the state variable, and the dynamics and the integral cost are allowed to have a discontinuous behaviour w.r.t. the time variable t in the following sense: although they have everywhere one-sided limits in t, they are required to be continuous only for a.e. t. For this class of problems we establish conditions under which the Value Function is characterized as the unique viscosity solution in the class of lower semicontinuous functions to the associated Hamilton-Jacobi equation. We provide some illustrative examples including a “growth versus consumption” problem in neo-classical macro-economics, one peculiarity of which is the presence of a fractional singularity w.r.t. the state variable.