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:
- $\map f 0 = 0$
Thus, for all $x \in \R$, we have:
\(\ds 0\) | \(=\) | \(\ds \map f 0\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map f {x + \paren {-x} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map f x + \map f {-x}\) |
It follows that the function $f$ is odd:
- $\forall x \in \R: \map f {-x} = -\map f x$.
$\blacksquare$