 Additive function

An additive function on an additive semigroup satisfies $f(x+y)=f(x)+f(y)$ whenever $x$, $y$, and $x+y$ belong to its domain.

For real numbers, an additive function is a group homomorphism from $(\mathbb{R},+)$ to itself. It preserves scalar multiplication by every rational number, so it is a linear map when domain and codomain are viewed as vector spaces over the rational numbers. Being a linear map over the real numbers additionally requires $f(tx)=tf(x)$ for every real number $t$. On the real line this stronger condition is equivalent to $f(x)=xf(1)$. Additivity alone does not imply it.