Our teacher told us to make three maps $\left ( U, φ \right ), \left ( V, ψ \right ), \left ( W, χ \right )$ in $\mathbb{R}$ such that $\left \{φ,ψ \right\}$ and $\left \{ψ,χ\right\}$ are differentiably compatible, but $\left \{φ,χ\right\}$ aren't.
Two maps $\left ( U, φ \right ), \left ( V, ψ \right )$ to be differentiably compatible means that the functions:
$$φ\circ ψ^{-1}: ψ(U\cap V)\rightarrow φ(U\cap V)$$ $$ψ\circ φ^{-1}: φ(U\cap V)\rightarrow ψ(U\cap V)$$ are $\mathcal{C^\infty}$, i.e. infinite times differentiable.
I can't see how this is going to happen, because when you compose two $\mathcal{C^\infty}$ functions the outcome is $\mathcal{C^\infty}$. Am I missing something? Do you have any idea how to proceed? Thank you very much!
Hint: Consider $\varphi : \mathbb R \to \mathbb R$ given by $\varphi(x) = x$ and $\psi: \left(\frac{\pi}{2}, \frac{3\pi}{2}\right) \to \mathbb R$, $\psi (x) = \tan x$, $\chi : \mathbb R \to \mathbb R$, $\chi (x) = x^3$.