Monotone Additive Function is Linear

Theorem
Let $f:\R\to\R$ be a monotonic function such that $f\left(x+y\right)=f\left(x\right)+f\left(y\right)$ for all $x,y\in\R$. Then, there exists $a$ such that $f\left(x\right)=ax$ for all $x\in\R$.

Proof
Let $a=f\left(1\right)$.

Then $f\left(1\right)=a\times1$. Supposing, by induction, that $f\left(n\right)=an$ for some $n\in\N$, we have $f\left(n+1\right)=f\left(n\right)+f\left(1\right)$ which is $an+a=a\left(n+1\right)$. So:


 * $\forall n\in\mathbb{N}, f\left(n\right)=an$