  We continue the work started in parts (I) and (II). In this part we classify
which continuous type A quivers are derived equivalent and introduce the new
continuous cluster category with E-clusters, which are a generalization of
clusters. In the middle we provide a rigorous connection between the previous
construction of the continuous cluster category and the new construction. We
conclude with the introduction of a cluster theory, generalizing the notion of
a cluster structure. Using this new notion, we demonstrate how one embeds known
type A cluster theories into the new E-cluster theory in a way compatible with
mutation. This is part (III) in a series of work that will conclude with a
continuous generalization of mutation for cluster theories.
