# Odd Function of Zero is Zero

## Theorem

Let $f: \R \to \R$ be an odd function.

Let $f$ be defined at the point $x = 0$.

Then:

$\map f 0 = 0$

## Proof

By definition of odd function:

$\map f {-x} = -\map f x$

and so:

 $\displaystyle \map f {-0}$ $=$ $\displaystyle \map f 0$ $\displaystyle$ $=$ $\displaystyle -\map f 0$

The only real number $a$ for which $a = -a$ is $0$.

Hence the result.

$\blacksquare$