In this study, the fundamental formulation for the continuous and discontinuous contact problems of a functionally graded (FG) layer lying on a rigid substrate are obtained without using superposition technique. The general expressions for the stresses and the displacements are derived in the presence of body forces using elasticity theory and Fourier integral transforms. As an application, the continuous and discontinuous contact problems of FG layer indented by a rigid cylindrical punch are considered. The contact problems are solved under the boundary conditions and the plane contact problems are reduced to the singular integral equations both in the case of continuous and discontinuous contact. The singular integral equations are solved numerically using Gauss-Chebychev integration formula and an iterative scheme is employed to obtain the correct contact width and critical load that satisfy the related conditions. Numerical results for critical load P-cr, initial separation distance, the separation region in the discontinuous contact, contact width under the punch, contact stress distributions between the layer-punch and the layer-substrate are given.