A rational function $f : A^* \to B^*$ is computed by a Mealy machine if and only if it is letter-to-letter — the output has the length of the input — and deterministic: input strings that agree on their first $n$ letters have outputs that agree on their first $n$ letters (Theorem B.2.7 of Transducers). Mealy machines are the special case of the rational functions that the theorem pins down; the proof goes through the machine-independent characterisation of Theorem B.4.1, since a rational function is continuous.