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

Result R2964 on D4692: Euclidean real function even part

Even function equals its even part

Formulation 0

Let $f : \mathbb{R}^n \to \mathbb{R}^m$ be an D3997: Even euclidean real function.
Let $f_{\mathsf{even}} : \mathbb{R}^n \to \mathbb{R}^m$ be the D4692: Euclidean real function even part of $f$.

Then \begin{equation} f = f_{\mathsf{even}} \end{equation}

Proofs

Proof 0

Let $f : \mathbb{R}^n \to \mathbb{R}^m$ be an D3997: Even euclidean real function.
Let $f_{\mathsf{even}} : \mathbb{R}^n \to \mathbb{R}^m$ be the D4692: Euclidean real function even part of $f$.

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