Monotone Additive Function is Linear

Theorem
Let $f: \R \to \R$ be a monotone real function which is additive, that is:
 * $\forall x, y \in \R: \map f {x + y} = \map f x + \map f y$

Then:
 * $\exists a \in \R: \forall x \in \R: \map f x = a x$

That is, $f$ is a linear function.