  We study the quasi-order of topological embeddability on definable functions
between Polish zero-dimensional spaces. We first study the descriptive
complexity of this quasi-order restricted to the space of continuous functions.
Our main result is the following dichotomy: the embeddability quasi-order
restricted to continuous functions from a given compact space to another is
either an analytic complete quasi-order or a well-quasi-order. We then turn to
the existence of maximal elements with respect to embeddability in a given
Baire class. It is proved that the class of continuous functions is the only
Baire class to admit a maximal element. We prove that no Baire class admits a
maximal element, except for the class of continuous functions which admits a
maximum element.
