Let $\Bbb R$ be the real line with the differentiable structure given by the maximal atlas of the chart
($\Bbb R,f = 1:\Bbb R→\Bbb R$), and let $\Bbb R′$ be the real line with the differentiable structure given by the
maximal atlas of the chart $(\Bbb R,y : \Bbb R→\Bbb R)$, where $y(x) = x \frac13$.
(a) Show that these two differentiable structures are distinct.
(b) Show that there is a diffeomorphism between $\Bbb R$ and $\Bbb R′$.
(Hint: The identity map $\Bbb R→\Bbb R$ is not the desired diffeomorphism; in fact, this map is not smooth.)
