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 Quote by Mathjunkie Reply to lurflurf: In the following n is an integer, x is a continuous variable (complex if you want). That is exactly the point, and it is a very important one that comes up many places - quantum field theory being a prime example. The business of generalizing a function over a new region of the independant variable is frought with danger, and much care is required. Your example (define sin(x) only when it is rational and say it is something else when it isn't) is an example of the ambiguity that exists. There is a theorem that says that a function can be UNIQUELY "analytically continued" from a region where the independent variable is continuous, no matter how small that region is. The rationals are NOT continuous; neither are the positive integers. Therefore, any function that is defined only over the integers (or rationals) can be extended (i.e. generalized, NOT "analytically continued" which is something very different) in an infinite number of ways. Simply setting n=x doesn't always work, although it is usually the best choice. Example: A(x) is any function you care to name that is finite when x=n: generalize as follows: x! = gamma(x+1) + A(x)*sin(Pi*x). Then n! will still equal gamma(n+1), but anyone who uses the usual definition will get a different result when x.ne.n. So, everyone has to come to some agreement, and the agreement is that x!=GAMMA(x+1). But the fact that someone had to ask the question suggests that not everyone knows the convention. So, if you find a textbook that refers to x!, somewhere in that book it should be written that x!=GAMMA(x+1). Otherwise the author is lazy, and worse, mathematically imprecise.
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.