Additive function

Summary

An 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. Between abelian groups it is the same thing as a group homomorphism. Every linear map is additive, and on the real line the continuous additive functions are exactly $x \mapsto cx$. With the axiom of choice there are also wild additive functions on $\R$ whose graphs are dense in the plane. In 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.