I What is *not* a Vector Function?

Depends on what exactly you call "vector function". If the codomain has to be a vector space, then simply take a set as codomain that cannot be interpreted as vector space:
f:{1} -> {1,2,3,4,5,6}, f(1)=1.

 

But here ##f## is basically equal to the linear embedding of ##\{0\}## into the two-dimensional vector space ##\mathbb{Z}_3^2##

 

That vector space has 9 elements, not 6. That number was carefully chosen.

 

Ooops ...