I recently became aware of the rather appealing characterization of deterministic word-to-word transductions as word functions with bounded variation (see e.g. ). This coincides with the set of functions that can be realized by subsequential transducers, which are deterministic finite state transducers with an output function defined on accepting states.
Not every function defined by a non-deterministic FST is subsequential, that is, even if the FST is single-valued, it might not be determinizable. I wondered if there are any extensional characterizations of the set of functions definable by (single-valued) FSTs?
 Bruyère, Véronique and Reutenauer, Christophe. A proof of choffrut's theorem on subsequential functions. Theoretical Computer Science, 1999.