I want familiar before with the concept of analytic continuations and so this part really confused me:

("The" is the appropriate article to use because the analytic continuation of a function is unique.)

Let f: N -> N, f(x) = x be a function on the natural numbers. Then I could define two functions g(x) and h(x) on N u {foo} that behave just like f for natural numbers. However, g(foo) is 42 while h(foo) is 666. Wouldn't g and h both be valid analytic continuations of f in N u {foo}, according to the definition of analytic continuations explained in the article?

I'm wondering about that as the uniqueness seems to be an important property for the rest of the explanation, yet it is simply assumed here without any further explanation.