Additive Function of Zero is Zero
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Theorem
Let $f: \R \to \R$ be an additive function.
Then:
- $\map f 0 = 0$
Proof
As $f$ is additive, we have:
\(\ds \map f 1\) | \(=\) | \(\ds \map f {0 + 1}\) | Real Addition Identity is Zero | |||||||||||
\(\ds \) | \(=\) | \(\ds \map f 0 + \map f 1\) | Definition of Additive Function |
that is:
- $\map f 0 = 0$
$\blacksquare$