Partial computability and regular computability are not mutually exclusive. In fact,
Theorem. Every computable language is partially computable.
Proof. Suppose that C is a computable language. Here is a program V that partially computes C.
V(x)
If x ∈ C then
stop and answer yes
else
loop forever
end if
The test x ∈ C can be done because C is computable. ◊