In a few places where I have looked the Euclidean Function of a Euclidean Domain is only being defined for nonzero elements. I am teaching an undergraduate course and I am trying to make things simpler as possible. Is there any good reason why not to define it as $0$ at $0$?
You'll find your answer and much more in the littleknown paper [1] which surveys all of the dozen known ways of axiomatizing Euclidean rings (including those of Nagata and Samuel), and explores indepth all of their logical interrelations. It's a convenient reference to have at hand when you're comparing texts which use (seemingly) different definitions of Euclidean rings / domains. [1] Euclidean Rings. A. G. Agargun, C. R. Fletcher 


I think this is only because you don't care dividing by zero. But you may as well define the value at zero to be 0, and any other value at least 1. See this expository paper by Keith Conrad for interesting remarks on euclidean domains and functions. 

