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  "body": "\n## Summary\n\nAn additive function is a function $f$ with $f(x + y) = f(x) + f(y)$ for all $x$ and $y$, that is, a solution of [[Cauchy's functional equation|ZBMEVWN100]].\nBetween abelian groups it is the same thing as a [[group homomorphism|0JYTG5K9Y4]].\nEvery linear map is additive, and on the real line the continuous additive functions are exactly $x \\mapsto cx$.\nWith the axiom of choice there are also wild additive functions on $\\R$ whose graphs are dense in the plane.\nIn number theory the phrase \"additive function\" has another meaning, for arithmetic functions with $f(mn) = f(m) + f(n)$ when $m$ and $n$ are coprime.\n\n## Explanations\n\n### L1 human\n\nAn additive function turns sums into sums: the output for $x + y$ is the output for $x$ plus the output for $y$.\nDoubling, $f(x) = 2x$, is additive, because $2(x + y) = 2x + 2y$.\nSquaring is not, because $(x + y)^2$ is not $x^2 + y^2$ in general.\nAdding a constant is not either: $f(x) = x + 1$ gives $f(0 + 0) = 1$ but $f(0) + f(0) = 2$.\n",
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