Problem
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?