  Given a continuous function $f:X\to\mathbb{R}$ and a cover $\mathcal{I}$ of
its image by intervals, the Mapper is the nerve of a refinement of the pullback
cover $f^{-1}(\mathcal{I})$. Despite its success in applications, little is
known about the structure and stability of this construction from a theoretical
point of view. As a pixelized version of the Reeb graph of $f$, it is expected
to capture a subset of its features (branches, holes), depending on how the
interval cover is positioned with respect to the critical values of the
function. Its stability should also depend on this positioning. We propose a
theoretical framework that relates the structure of the Mapper to the one of
the Reeb graph, making it possible to predict which features will be present
and which will be absent in the Mapper given the function and the cover, and
for each feature, to quantify its degree of (in-)stability. Using this
framework, we can derive guarantees on the structure of the Mapper, on its
stability, and on its convergence to the Reeb graph as the granularity of the
cover $\mathcal{I}$ goes to zero.
