The continuous interior penalty (CIP) method for elliptic convection-diffusion problems with characteristic layers on a Shishkin mesh is analysed. The method penalises jumps of the normal derivative across interior edges. We show that it is of the same order of convergence as the streamline diffusion finite-element method and is superclose in the CIP norm induced by its bilinear form for the difference between the FEM solution and the bilinear nodal interpolant of the exact solution. Furthermore, we study numerically the behaviour of the method for different choices of the stabilisation parameter.