  The relative interlevel set cohomology (RISC) is an invariant of real-valued
continuous functions closely related to the Mayer--Vietoris pyramid introduced
by Carlsson, de Silva, and Morozov. As such, the relative interlevel set
cohomology is a parametrization of the cohomology vector spaces of all open
interlevel sets relative complements of closed interlevel sets. We provide a
structure theorem, which applies to the RISC of real-valued continuous
functions whose open interlevel sets have finite-dimensional cohomology in each
degree. Moreover, we show this tameness assumption is in some sense equivalent
to $q$-tameness as introduced by Chazal, de Silva, Glisse, and Oudot.
Furthermore, we provide the notion of an interleaving for RISC and we show that
it is stable in the sense that any space with two functions that are
$\delta$-close induces a $\delta$-interleaving of the corresponding relative
interlevel set cohomologies. Finally, we provide an elementary form of
quantitative homotopy invariance for RISC.
