ThmDex – An index of mathematical definitions, results, and conjectures.

Result R2960 on D3998: Odd euclidean real function

Function odd part is odd

Formulation 0

Let $f : \mathbb{R}^n \to \mathbb{R}^m$ be a D992: Function.
Let $f_{\mathsf{odd}} : \mathbb{R}^n \to \mathbb{R}^m$ be the D4693: Function odd part of $f$.

Then $f_{\mathsf{odd}}$ is an D3998: Odd euclidean real function.

Proofs

Proof 0

Let $f : \mathbb{R}^n \to \mathbb{R}^m$ be a D992: Function.
Let $f_{\mathsf{odd}} : \mathbb{R}^n \to \mathbb{R}^m$ be the D4693: Function odd part of $f$.

If $x \in \mathbb{R}^n$, then \begin{equation} f_{\mathsf{odd}}(-x) = \frac{f(-x) - f(x)}{2} = - \frac{f(x) - f(- x)}{2} = - f_{\mathsf{odd}}(x) \end{equation} $\square$