  A uniformizer of a binary relation is a function whose graph is contained in
the relation and which is defined on the same domain as the relation. It is
known that any rational relation of infinite words, i.e. a relation given as a
transducer, admits a rational uniformizer. Although rational, those
uniformizers are not necessarily well-behaved, in the sense that the $i$th
letter of the output word may depend on the whole infinite input word. In other
words, those uniformizers might not be continuous (for the Cantor topology).
This paper addresses the question of whether rational relations of infinite
words can be uniformized by continuous functions. On the negative side,
continuous uniformizers might not exist in general and we prove that deciding
their existence is algorithmically impossible. On the positive side, we exhibit
a large class of rational relations of infinite words, called weakly
deterministic rational relations, for which deciding whether a relation in this
class admits a continuous uniformizer is an ExpTime-c problem. This class
includes the known classes of deterministic rational relations and automatic
relations of infinite words.
  As an application of the previous result, and by exploiting a connection
between computability and continuity for rational functions of infinite words,
we show a result on the synthesis of computable functions from specifications
given as weakly deterministic rational relations. In particular, we show that
deciding the existence of a computable uniformizer is ExpTime-c and if there is
one, it is possible to effectively synthesize a deterministic two-way
transducer computing it. This generalizes the classical setting of Church
synthesis to asynchronous implementations which can arbitrarily delay the
production of their output signals.
