The paper presents a technique for modeling arbitrary discontinuities in finite elements, including discontinuities in the function and its derivatives. The method uses signed distance functions to define the surfaces of discontinuity, allowing for the use of level sets to update the position of the discontinuities. The discontinuous approximation is constructed using shape functions and enrichment functions, which are applied to nodes that are bisected or intersected by the discontinuity. The paper also discusses the weak form for Laplace equations with interior discontinuities and equilibrium equations with discontinuous motion. Numerical examples are provided to illustrate the method's effectiveness in modeling crack propagation, journal bearings, circular inclusions under compression, and jointed rock masses. The method is particularly useful for problems with stationary and evolving discontinuities without the need for remeshing.The paper presents a technique for modeling arbitrary discontinuities in finite elements, including discontinuities in the function and its derivatives. The method uses signed distance functions to define the surfaces of discontinuity, allowing for the use of level sets to update the position of the discontinuities. The discontinuous approximation is constructed using shape functions and enrichment functions, which are applied to nodes that are bisected or intersected by the discontinuity. The paper also discusses the weak form for Laplace equations with interior discontinuities and equilibrium equations with discontinuous motion. Numerical examples are provided to illustrate the method's effectiveness in modeling crack propagation, journal bearings, circular inclusions under compression, and jointed rock masses. The method is particularly useful for problems with stationary and evolving discontinuities without the need for remeshing.