The recently introduced Fragile Points Method (FPM) is extended to study the static bending, free vibration and eigen buckling of functionally graded beams. The beam kinematics is based on Euler-Bernoulli theory. The salient feature of the FPM is that it is a truly meshless method that employs simple local point-based polynomial test and trial functions, unlike element-based trial and test functions. The distinguishing feature is that, unlike the traditional Galerkin framework, the polynomial test and trial functions are discontinuous (piece-wise continuous over the global domain) that are derived by employing the generalized finite difference method. Further, as the trial and test functions are discontinuous, the continuity requirement imposed by the continuous Galerkin framework is circumvented. The discontinuous trial and test functions lead to inconsistency, to alleviate this, we employ numerical flux corrections inspired by the discontinuous Galerkin method. The efficiency and robustness of the approach are demonstrated with a few standard benchmark examples.