Problem
Problem 1.22 Prove that the function $\varphi: \mathbb{R} \rightarrow \mathbb{R}, \varphi(s)=s^{3}$, defines a $C^{\infty}$ differentiable structure on $\mathbb{R}$ different from the usual one (that of the atlas $\left.\left\{\left(\mathbb{R}, \operatorname{id}_{\mathbb{R}}\right)\right\}\right)$.