This paper presents a novel method to treat discontinuities in a 3D piece-wise non-rigid registration framework, coined as EXtended Free-Form Deformation (XFFD). Existing discontinuities in the image, such as sliding motion of the lungs or the cardiac boundary adjacent to the blood pool, should be handled to obtain physically plausible deformation fields for motion analysis. However, conventional Free-form deformations (FFDs) impose continuity over the whole image, introducing inaccuracy near discontinuity boundaries. The proposed method incorporates enrichment functions into the FFD formalism, inspired by the linear interpolation method in the EXtended Finite Element Method (XFEM). Enrichment functions enable B-splines to handle discontinuities with minimal increase of computational complexity, while avoiding boundary-matching problem. It retains all properties of the framework of FFDs yet seamlessly handles general discontinuities and can also coexist with other proposed improvements of the FFD formalism. The proposed method showed high performance on synthetic and 3D lung CT images. The target registration error on the CT images is comparable to the previous methods, while being a generic method without assuming any type of motion constraint. Therefore, it does not include any penalty term. However, any of these terms could be included to achieve higher accuracy for specific applications.