What property of a function allows you to split function applications across a summation?

Question

Say I have a function $f(x)$ such that $x$ is a natural number and $f(x)$ is a natural number, if I can prove that there exists natural numbers $x$ and $y$ such that $$f(x+y) \neq f(x) + f(y)$$ Am I able to say this disproves a certain type of rule of replacement for the function? Is there any other property of the function that is implied by this inequality?

Answer 1

With thanks to @lulu above, it seems the exact property of the function I am thinking of is that it does not "preserve addition" (a property of linear functions). So by proving there exists an $x$ and $y$ such that addition is not preserved, we can at least say that the function is nonlinear.