We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$ Deduce in the general case that, if $\xi : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function satisfying condition (V.1), then it is odd and its restriction to $\mathbb{R}_+^*$ is of the form $x \mapsto Cx^\beta$, with $C \neq 0$ and $\beta > 0$.
We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where
$$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$
Deduce in the general case that, if $\xi : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function satisfying condition (V.1), then it is odd and its restriction to $\mathbb{R}_+^*$ is of the form $x \mapsto Cx^\beta$, with $C \neq 0$ and $\beta > 0$.