We are given a function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ and we define a function $f_\xi : \mathcal{M}_d(\mathbb{R}) \rightarrow \mathcal{M}_d(\mathbb{R})$ such that $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad f_\xi(A) = \left(\xi\left(A_{i,j}\right)\right)_{1 \leqslant i,j \leqslant d}$$ We propose to determine the continuous functions $\xi : \mathbb{R} \rightarrow \mathbb{R}$ such that $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) \text{ invertible} \tag{V.1}$$ Determine the continuous functions $\xi$ satisfying condition (V.1) when $d = 1$.
We are given a function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ and we define a function $f_\xi : \mathcal{M}_d(\mathbb{R}) \rightarrow \mathcal{M}_d(\mathbb{R})$ such that
$$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad f_\xi(A) = \left(\xi\left(A_{i,j}\right)\right)_{1 \leqslant i,j \leqslant d}$$
We propose to determine the continuous functions $\xi : \mathbb{R} \rightarrow \mathbb{R}$ such that
$$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) \text{ invertible} \tag{V.1}$$
Determine the continuous functions $\xi$ satisfying condition (V.1) when $d = 1$.