Abstract:  Dominance reasoning is a paradigm of rationality.  Intransitive preferences are a paradigm of irrationality.  Yet there are cases where an infinite sequence of decisions guided by dominance reasoning leads to a situation closely analogous to having intransitive preferences.  In these cases, the agent has utilities that are ‘discontinuous at infinity’.  I describe the structure of three such cases, explain what we might mean by discontinuity at infinity and why it poses a challenge for decision theory, and explore several possible responses to the challenge.