Additive Function is Odd Function
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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:
\(\displaystyle 0\) | \(=\) | \(\displaystyle f \paren 0\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle f \paren {x + \paren {-x} }\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle f \paren x + f \paren {-x}\) |
It follows that the function $f$ is odd:
- $\forall x \in \R: f \paren {-x} = -f \paren x$.
$\blacksquare$