Additive Function of Zero is Zero

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:

$\map f 0 = 0$

$\blacksquare$