Is there any good literature on the computational complexity of function problems?
There is an article from Alan Selman: A taxonomy of complexity classes of functions

Please note, that i found this reference in an Answer (by Joshua Grochow) for this Question:

Complexity class when reducing decision problem to function problem

Note also the Comments from Kaveh:


  If you have a function you can look at the bit graph of the function which is a decision problem. If you can compute the function you can compute the bit graph easily from it. If you can compute the bit graph you can compute the function by simple computing the bits of the output one by one.


if i found more, i will add it here.