  A word-to-word function is rational if it can be realized by a
non-deterministic one-way transducer. Over finite words, it is a classical
result that any rational function is regular, i.e. it can be computed by a
deterministic two-way transducer, or equivalently, by a deterministic streaming
string transducer (a one-way automaton which manipulates string registers).
  This result no longer holds for infinite words, since a non-deterministic
one-way transducer can guess, and check along its run, properties such as
infinitely many occurrences of some pattern, which is impossible for a
deterministic machine. In this paper, we identify the class of rational
functions over infinite words which are also computable by a deterministic
two-way transducer. It coincides with the class of rational functions which are
continuous, and this property can thus be decided. This solves an open question
raised in a previous paper of Dave et al.
