Summary: We prove that discontinuous solutions of the Navier-Stokes equations for one-dimensional, compressible fluid flow depend continuously on their initial data. Perturbations in the different components are measured in various fractional Sobolev norms; \(L^2\) bounds are then obtained by interpolation. This improves upon earlier results in which continuous dependence was known only in a much stronger topology, one inappropriately strong for the physical model. More generally, we derive a bound for the difference between exact and approximate weak solutions in terms of their initial differences and of the weak truncation error associated with the approximate solution.