How did quantum field theory or analytically contination get into this? It seemed as if you meant it was (bad) lazy to make the definition x! = gamma(x+1). If you only meant the definition should be clearly stated, that is fine. Though it is tedious to clearly define everything at all times. There is really not a competing definition of x!, x! = gamma(x+1) + A(x)*sin(Pi*x) having not caught on. No confusion results. As I already stated it is desirable that x! be log convex. Speaking of being (bad) lazy one does not normally speak of sets like N, Q, and R as being continuous or not. When one speaks of a function being continuous, one specifies a topology or equivelently type of limit that defines the continuity.
If we define f(x)=sin(x*pi) when x is rational (that is Q is the domain of f)
Theorem 1: f(x) is (rational) continuous for all x
because sin(x+h)-sin(x) is defined and small for all x and small h (where x+h and x are in the domain of f)
Theorem 2: f(x) is an algebraic number for all x
So Theorem 1 will remain true, but theorem 2 will be ruined by the generalization to real numbers.