Yes, there are no possible return values. But there is a unique vacuous total function because no return values are ever needed. A function can be considered as

Yes, there are no possible return values. But there is a unique vacuous total function because no return values are ever needed. A function can be considered as a set of ordered pairs, one for each value in the domain. Since there are no values in the domain, no pairs need exist. Another way to look at it is as a constrained subset of the Cartesian product of domain and codomain. In this case the domain is the empty set, so the Cartesian product is empty too.