We study the quasi-order of topological embeddability on definable functions between Polish
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-dimensional spaces.
We consider 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 also investigate the existence of maximal elements with respect to embeddability in a given Baire class. We prove that no Baire class admits a maximal element, except for the class of continuous functions which admits a maximum element.