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