Manifolds Part 1

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 $\left.\left\{\left(\mathbb{R}, \operatorname{id}_{\mathbb{R}}\right)\right\}\right)$.

Problem 1.23 Prove that if $h: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a homeomorphism, then the atlas $\left\{\left(\mathbb{R}^{n}, h\right)\right\}$ defines the usual differentiable structure on $\mathbb{R}^{n}$ (that defined by the atlas $\left.\left\{\left(\mathbb{R}^{n}, \operatorname{id}_{\mathbb{R}^{n}}\right)\right\}\right)$ if and only if $h$ and $h^{-1}$ are differentiable.

Problem 1.24 For each real number $r>0$, consider the map $\varphi_{r}: \mathbb{R} \rightarrow \mathbb{R}$, where $\varphi_{r}(t)=t$ if $t \leqslant 0$ and $\varphi_{r}(t)=r t$ if $t \geqslant 0$. Prove that the atlases $\left\{\left(\mathbb{R}, \varphi_{r}\right)\right\}_{r>0}$ define an uncountable family of differentiable structures on $\mathbb{R}$. Are the corresponding differentiable manifolds diffeomorphic?