  A word-to-word function is continuous for a class of languages~$\mathcal{V}$
if its inverse maps $\mathcal{V}$_languages to~$\mathcal{V}$. This notion
provides a basis for an algebraic study of transducers, and was integral to the
characterization of the sequential transducers computable in some circuit
complexity classes.
  Here, we report on the decidability of continuity for functional transducers
and some standard classes of regular languages. To this end, we develop a
robust theory rooted in the standard profinite analysis of regular languages.
  Since previous algebraic studies of transducers have focused on the sole
structure of the underlying input automaton, we also compare the two algebraic
approaches. We focus on two questions: When are the automaton structure and the
continuity properties related, and when does continuity propagate to
superclasses?
