Let \( \varphi : \mathbb{R} \to \mathbb{R} \) be given by
\[ \varphi(x) = x^3. \]The atlas \( (\mathbb{R}, \varphi) \) is smooth since it consists of only one chart. Thus, it determines a smooth structure on \( \mathbb{R} \) (see (0x68f0e02a) )
Remarks
- Since \[ \mathrm{id}_{\mathbb{R}} \circ \varphi^{-1}(y) = y^{1/3} \] is not smooth at the origin, the maps \( \mathrm{id}_{\mathbb{R}} \) and \( \varphi \) are not compatible. Hence, the structure determined by \( (\mathbb{R}, \varphi) \) is distinct from the standard one . However, both are diffeomorphic to each other.