We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), and $\eta = \xi^{-1} : I \rightarrow \mathbb{R}$ is the inverse function of the bijection $\xi : \mathbb{R} \rightarrow I$.
We assume in this question that the function $\eta$ takes strictly positive values on $I \cap {]0, +\infty[}$.
1) Show that the function $f = \ln \circ \eta \circ \exp$ satisfies equation (IV.1) on an interval $]-\infty, M[$, with $M$ (possibly infinite) to be determined as a function of the interval $I$.
2) Deduce that on the interval $I \cap {]0, +\infty[}$ the function $\eta$ is of the form
$$\eta : x \mapsto K_1 x^{\alpha_1}$$
with two constants $K_1 > 0$ and $\alpha_1 > 0$.
3) Show that on the interval $I \cap {]-\infty, 0[}$ the function $\eta$ is of the form
$$\eta : x \mapsto K_2(-x)^{\alpha_2}$$
with two constants $K_2 < 0$ and $\alpha_2 > 0$.
4) Show that $I = \mathbb{R}$ then that the function $\eta$ is an odd function.