This paper is concerned with the continuous and discontinuous contact problem of two elastic layers resting on an elastic semi-infinite plane. The top layer is subjected to a uniform pressure applied over a finite portion of its top surface. It is assumed that the contact between all surfaces is frictionless. The problem is solved using the theory of elasticity, and body forces are taken into account. Separation may occur between the top and the bottom layers or between the bottom layer and the half-plane or between both interfaces. The problem is formulated in terms of singular integral equations obtained from the discontinuous contact positions and is numerically solved by the Gauss-Chebyshev integration method. Furthermore, numerical results for the separations and the loads corresponding to these separations and the stress distribution on the contact interfaces are given in graphical forms.