  Let $H$ be a bounded and Lipschitz continuous function. We consider
discontinuous viscosity solutions of the Hamilton-Jacobi equation
$U_{t}+H(U_x)=0$ and signed Radon measure valued entropy solutions of the
conservation law $u_{t}+[H(u)]_x=0$. After having proved a precise statement of
the formal relation $U_x=u$, we establish estimates for the (strictly
positive!) times at which singularities of the solutions disappear. Here
singularities are jump discontinuities in case of the Hamilton-Jacobi equation
and signed singular measures in case of the conservation law.
