  We generalize type $A$ quivers to continuous type $A$ quivers and prove
initial results about pointwise finite-dimensional (pwf) representations. We
classify the indecomosable pwf representations and provide a decomposition
theorem, recovering results of Botnan and Crawley-Boevey. We also classify the
indecomposable pwf projective representations. Finally, we prove that many of
the properties of finite-dimensional type $A_n$ representations are present in
finitely generated pwf representations. This is the self-contained foundational
part of a series of works to study a generalization of continuous clusters
categories and their relationship to other type $A$ cluster structures.

Representation Theory