In this paper we present an error estimate for the explicit Runge-Kutta discontinuous Galerkin method to solve linear hyperbolic equation in one dimension with discontinuous but piecewise smooth initial data. The discontinuous ¯finite element space is made up of piecewise polynomials of arbitrary degree, and time is advanced by the third order explicit total variation diminishing Runge-Kutta method under the standard CFL temporal-spatial condition. The error at the ¯final time T in the L2(RnRT )-norm is the optimal order both in space and in time, where RT is the pollution region due to the initial discontinuity with the width of order O(h1=2 log(1=h)), where h is the maximum cell length.