  We discuss topological equicontinuity and even continuity in dynamical
systems. In doing so we provide a classification of topologically transitive
dynamical systems in terms of equicontinuity pairs, give a generalisation of
the Auslander-Yorke Dichotomy for minimal systems and show there exists a
transitive system with an even continuity pair but no equicontinuity point. We
define what it means for a system to be eventually sensitive; we give a
dichotomy for transitive dynamical systems in relation to eventual sensitivity.
Along the way we define a property called splitting and discuss its relation to
some existing notions of chaos.
