  We introduce stability conditions (in the sense of King) for representable
modules of continuous quivers of type A along with a special criteria called
the four point condition. The stability conditions are defined using a
generalization of delta functions, called half-delta functions. We show that
for a continuous quiver of type A with finitely many sinks and sources, the
stability conditions satisfying the four point condition are in bijection with
measured laminations of the hyperbolic plane. Along the way, we extend an
earlier result by the first author and Todorov regarding continuous cluster
categories for linear continuous quivers of type A and laminations of the
hyperbolic plane to all continuous quivers of type A with finitely many sinks
and sources. We also give a formula for the continuous cluster character.
