It is shown that the rate of convergence of numerical approximations to the solution of hyperbolic partial differential equations is degraded when the solution being sought is not sufficiently smooth. A pth-order finite-difference scheme may give worse than pth-order convergence if the ( p + 1)st derivatives of the solution are not piecewise continuous. Spectral methods, which are normally expected to give infinite-order convergence, give only finite-order convergence if some derivative of the solution is not continuous. Near a discontinuity propagating along a characteristic of the differential equation, the truncation error of difference approximations is much larger on one side than on the other, and it oscillates in sign on the side where it is larger. (It may also oscillate on the other side of the discontinuity, depending on the order of the numerical scheme used.) Finally our analysis shows that, even if the true solution is not smooth, high-order schemes are more accurate than lower-order schemes.