  We show that the quotient of the continuous cluster category $\mathcal C_\pi$
modulo the additive subcategory generated by any cluster is an abelian category
and we show that it is isomorphic to the category of infinite length modules
over the endomorphism ring of the cluster. These theorems extend the theorems
of Caldero-Chapoton-Schiffler and Buan-Marsh-Reiten for cluster categories to
the continuous cluster category of type $A$. These results will be generalized
in a series of forthcoming joint papers of the two authors with Job Rock.
