This paper presents the exact closed-form solutions for the stress fields induced by a two-dimensional (2D) non-uniform displacement discontinuity (DD) of finite length in an isotropic elastic half plane. The relative displacement across the DD varies quadratically. We employ the complex potential-function method to first determine the Green's stress fields induced by a concentrated force and then apply Betti's reciprocal theorem to obtain the Green's displacement fields due to concentrated DD. By taking the derivative of the Green's functions and integrating along the DD, we derive the exact closed-form solutions of the stress fields for a quadratic DD. The solutions are applied to analyze the stress fields near a horizontal DD in the half plane with three different profiles: uniform (constant), linear, and quadratic. The same methodology is applied to an inclined normal fault to investigate the effect of different DD profiles on the maximum shear stress in the half plane as well as on the normal and shear stresses along the fault. Numerical results demonstrate considerable influence of the DD profile on the stress distribution around the discontinuity.