= 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.
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