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:

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

$\blacksquare$