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?
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1$\begingroup$ Not clear what you are asking. Linear functions satisfy this property, but of course most functions do not. Does that answer your question? $\endgroup$– luluApr 13, 2022 at 20:23
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$\begingroup$ I think that does satisfy my question, the exact words used to describe this property had escaped me. $\endgroup$– brubsbyApr 13, 2022 at 20:30
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1$\begingroup$ To be clear, though, "linearity" mean more than just this. A linear function (in an appropriate context) satisfies this but it also satisfies $f(cx)=cf(x)$. This doesn't really make sense in the context of functions from $\mathbb N\to \mathbb N$ (or at least it's not very interesting since in this case it follows from the other property). $\endgroup$– luluApr 13, 2022 at 20:33
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$\begingroup$ Understood! The function I am attempting to describe fails to preserve addition for some inputs, so it seems to follow that it is nonlinear without worrying about whether it preserves scalar multiplication. The function preserves addition for some inputs, so I wanted to be able to describe the property of "preserving addition" by name to better reason about it. $\endgroup$– brubsbyApr 13, 2022 at 20:45
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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.