Decomposing Borel functions and structure at finite levels of the Baire hierarchy

We prove that if f is a partial Borel function from one Polish space to another, then either f can be decomposed into countably many partial continuous functions, or else f contains the countable infinite power of a bijection that maps a convergent sequence together with its limit onto a discrete space. This is a generalization of a dichotomy discovered by Solecki for Baire class 1 functions. As an application, we provide a characterization of functions which are countable unions of continuous functions with domains of type , for a fixed n<ω. For Baire class 1 functions, this generalizes analogous characterizations proved by Jayne and Rogers for n=1 and Semmes for n=2.