For some simple type of boundary conditions, errors introduced by the use of finite-difference representations in the region of a discontinuity, either in the initial conditions or between the initial conditions and the boundary conditions, may persist right through to the steady-state solution of a one-dimensional parabolic partial differential equations. These errors depend on the mesh length, δx, in the x direction, and, in the simplest cases, are equal to k(δx)p, where k is a constant depending on the initial distribution, and where the discontinuity occurs in the (p − 1)th derivative. Analytic expressions for these persistent discretization errors are given in simple cases, with a general initial distribution defined by its values at discrete points, δx apart. The error persist when the steady-state solution depends on the initial condition as well as the boundary conditions.