  Regular functions of infinite words are (partial) functions realized by
deterministic two-way transducers with infinite look-ahead. Equivalently, Alur
et. al. have shown that they correspond to functions realized by deterministic
Muller streaming string transducers, and to functions defined by
MSO-transductions. Regular functions are however not computable in general (for
a classical extension of Turing computability to infinite inputs), and we
consider in this paper the class of deterministic regular functions of infinite
words, realized by deterministic two-way transducers without look-ahead. We
prove that it is a well-behaved class of functions: they are computable, closed
under composition, characterized by the guarded fragment of MSO-transductions,
by deterministic B\"uchi streaming string transducers, by deterministic two-way
transducers with finite look-ahead, and by finite compositions of sequential
functions and one fixed basic function called map-copy-reverse.
