Let f: Z → Z be a function such that f(a + b) = f(a) + f(b) for all a,b ε Z. Prove that there exists an integer n such that f(a) = an for all a ε Z.
The Attempt at a Solution
I'm a little bit confused here if this is just supposed to be really simple, or if there's more to it that I'm just completely missing.
What I am thinking right now is that, couldn't n be any integer and you can just define f as a function f(x) = xn and therefore f(a + b) = (a + b)n = an + bn = f(a) + f(b) which satisfies the original statement?