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 Poisson’ ratio and exponentially varying shear modules and density. For continuous contact, the problem is solved analytically using plane elasticity and integral transform techniques. The critical load that causes first separation for various material properties is investigated.  The problem is reduced to a singular integral equation using plane elasticity and integral transform techniques in case of discontinuous contact. 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 between the FG layer and the foundation is analyzed. The results are shown in tables and figures. It is seen that decreasing stiffness and density at the top of the layer results an increment  in critical load and the lowest pressure occurs on the symmetry axis in case of continuous contact. In addition, the separation distance increases with decreasing stiffness and density at the top of the layer in case of discontinuous contact.