We consider the simplest example of a time-dependent first order Hamilton-Jacobi equation, in one space dimension and with a bounded and Lipschitz continuous Hamiltonian which only depends on the spatial derivative. We show that if the initial function has a finite number of jump discontinuities, the corresponding discontinuous viscosity solution of the corresponding Cauchy problem on the real line is unique. Uniqueness follows from a comparison theorem for semicontinuous viscosity sub-and supersolutions, using the barrier effect of spatial discontinuities of a solution. We also prove an existence theorem, as well as a comparison theorem for viscosity solutions with different initial data. In addition, we describe several properties of the evolution of the jump discontinuities.As a byproduct of our analysis, we obtain new existence and uniqueness results for the initial-boundary value problem on an interval with (possibly singular) Neumann boundary conditions.