  By definition, transverse intersections are stable under infinitesimal
perturbations. Using persistent homology, we extend this notion to a measure.
Given a space of perturbations, we assign to each homology class of the
intersection its robustness, the magnitude of a perturbations in this space
necessary to kill it, and prove that robustness is stable. Among the
applications of this result is a stable notion of robustness for fixed points
of continuous mappings and a statement of stability for contours of smooth
mappings.
