>>11071841I think I misinterpreted your question, I took it for you to ask why a given function can be not continued in another way.
(
To that I had already written this simpler version:
Let's try try it hands on: Say you have the function
p : R -> R
p(x):= x^5
Along the real number line R, the rate of change of p is given as
p'(x) = 5 x^4
and if you want to know a close value to any x, also in R, i.e. horizonally next to x, then
p(x+d) = p(x) + d * p'(x) + O(x^2)
Now say you want to analytically (emphasis on this property) continue p into the complex plane, i.e. want to move up vertically. Then
p(x+id) = p(x) + i * d * p'(x) + O(x^2)
and in p', the complex derivative tells you how to change both (two) components, even if you just started with a real function.
So the rate of change is fixed in either direction.
)
But in case you only say that there are more functions that reduce to something at a certain point, then sure.
If Z(s) is the Zeta function such that Z(-1) "is the sum of all naturals", then of course the function
U(s) := Z(s^15)
also has
U(-1) = Z(-1).
But this has nothing to do with analytic continuation.
