Odd Function of Zero is Zero

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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.

$\blacksquare$


Sources