# Additive Function is Odd Function

## Theorem

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

Then $f$ is an odd function.

## Proof

$\map f 0 = 0$

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

 $\displaystyle 0$ $=$ $\displaystyle \map f 0$ $\displaystyle$ $=$ $\displaystyle \map f {x + \paren {-x} }$ $\displaystyle$ $=$ $\displaystyle \map f x + \map f {-x}$

It follows that the function $f$ is odd:

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

$\blacksquare$