This paper seeks to generate a continuous pseudoinverse of a function that maps a higher dimensional compact set to a lower dimensional one. Continuity and smoothness should be attained if possible, but otherwise the volume of the discontinuity boundary should be minimized. A sampling-based approximation technique is presented that uses discretized roadmaps of both the domain and image, and minimizes discontinuities of the inverse function. The method is applied to kinematic redundancy resolution for redundant robots, which have more degrees of freedom than workspace dimensions. The output is a global redundancy resolution, which has the convenient property that whenever the robot returns to the same workspace point, it uses the same joint-space pose. If a global resolution cannot be found, then the method minimizes discontinuities and maps them in workspace. Results are demonstrated on toy problems with up to 20 DOF, and on several robot arms.