Additive Function is Odd Function

Theorem
Let $f: \R \to \R$ be an additive function.

Then $f$ is an odd function.

Proof
From Additive Function of Zero is Zero:
 * $f \paren 0 = 0$

Thus, for all $x \in \R$, we have:

It follows that the function $f$ is odd:


 * $\forall x \in \R: f \paren {-x} = -f \paren x$.