Before revision:  We also devise algorithms for the computation of this invariantand we furthermore show that the persistent cup-length is 2-Lipschitz continuous with respect to the homotopy interleaving and Gromov-Hausdorff distances .
After revision:  We also devise algorithms for the computation of this invariantand we furthermore show that this invariant is stable under suitable interleaving-type distances. Along the way, we identify an invariant which we call the Cup-Length Diagram, which is stronger than persistent cup-length is 2-Lipschitz continuous with respect to the homotopy interleaving and Gromov-Hausdorff distances .