  Given a function $f: \Xspace \to \Rspace$ on a topological space, we consider
the preimages of intervals and their homology groups and show how to read the
ranks of these groups from the extended persistence diagram of $f$. In
addition, we quantify the robustness of the homology classes under
perturbations of $f$ using well groups, and we show how to read the ranks of
these groups from the same extended persistence diagram. The special case
$\Xspace = \Rspace^3$ has ramifications in the fields of medical imaging and
scientific visualization.
