In this study, the continuous and discontinuous contact problem of a functionally graded (FG) layer resting on a rigid foundation is considered. The top of the FG layer is subjected to normal tractions over a finite segment. The graded layer is modeled as a non-homogenous medium with a constant Poissons' ratio and exponentially varying shear modules and density. For continuous contact, the problem was solved analytically using plane elasticity and integral transform techniques. The critical load that causes first separation and contact pressures is investigated for various material properties and loadings. The problem reduced to a singular integral equation using plane elasticity and integral transform techniques in case of discontinuous contact. The obtained singular integral equation is solved numerically using Gauss-Jacobi integral formulation, and an iterative scheme is employed to obtain the correct separation distance. The separation distance and contact pressures between the FG layer and the foundation are analyzed for various material properties and loading. The results are shown in Tables and Figures. It is seen that decreasing stiffness and density at the top of the layer result in an increment in both critical load in case of continuous contact and separation distance in case of discontinuous contact.