Problem
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.