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There are infinitely many differentiable structures on $\mathbb R$ : take any homeomorphism which is no diffeomorphism (such as $x\mapsto x^3$), and you get an non-usual differentiable structure on $\mathbb R$!

Even better : there exist uncountably many different real analytic structures on $\mathbb R$. But this example is general : for every $k\in \mathbb N \cup \{\infty,\omega\}$, the group Homeo($\mathbb R$) acts transitively on the set of $C^k$ differentiable structures on $\mathbb R$.

EDIT : If you want to show that up to $\textit{diffeomorphism of differentiable manifold}$, there exists only one differentiable structure on $\mathbb R$, just remember that (more generally) all $1$-dimensional non-compact manifolds are diffeomorphic to $\mathbb R$. [this is proved in Lafontaine e.g]

There are infinitely many differentiable structures on $\mathbb R$ : take any homeomorphism which is no diffeomorphism (such as $x\mapsto x^3$), and you get an non-usual differentiable structure on $\mathbb R$!

Even better : there exist uncountably many different real analytic structures on $\mathbb R$. But this example is general : for every $k\in \mathbb N \cup \{\infty,\omega\}$, the group Homeo($\mathbb R$) acts transitively on the set of $C^k$ differentiable structures on $\mathbb R$.

EDIT : If you want to show that up to diffeomorphism of differentiable manifold, there exists only one differentiable structure on $\mathbb R$, just remember that (more generally) all $1$-dimensional non-compact manifolds are diffeomorphic to $\mathbb R$. [this is proved in Lafontaine e.g]

There are infinitely many differentiable structures on $\mathbb R$ : take any homeomorphism which is no diffeomorphism (such as $x\mapsto x^3$), and you get an non-usual differentiable structure on $\mathbb R$!

Even better : there exist uncountably many different real analytic structures on $\mathbb R$. But this example is general : for every $k\in \mathbb N \cup \{\infty,\omega\}$, the group Homeo($\mathbb R$) acts transitively on the set of $C^k$ differentiable structures on $\mathbb R$.

EDIT : If you want to show that up to homeomorphism, there exists only one differentiable structure on $\mathbb R$, just remember that (more generally) all $1$-dimensional non-compact manifolds are diffeomorphic to $\mathbb R$. [this is proved in Lafontaine e.g]

There are infinitely many differentiable structures on $\mathbb R$ : take any homeomorphism which is no diffeomorphism (such as $x\mapsto x^3$), and you get an non-usual differentiable structure on $\mathbb R$!

Even better : there exist uncountably many different real analytic structures on $\mathbb R$. But this example is general : for every $k\in \mathbb N \cup \{\infty,\omega\}$, the group Homeo($\mathbb R$) acts transitively on the set of $C^k$ differentiable structures on $\mathbb R$.

EDIT : If you want to show that up to $\textit{diffeomorphism of differentiable manifold}$, there exists only one differentiable structure on $\mathbb R$, just remember that (more generally) all $1$-dimensional non-compact manifolds are diffeomorphic to $\mathbb R$. [this is proved in Lafontaine e.g]

There are infinitely many differentiable structures on $\mathbb R$ : take any homeomorphism which is no diffeomorphism (such as $x\mapsto x^3$), and you get an non-usual differentiable structure on $\mathbb R$!

Even better : there exist uncountably many different real analytic structures on $\mathbb R$. But this example is general : for every $k\in \mathbb N \cup \{\infty,\omega\}$, the group Homeo($\mathbb R$) acts transitively on the set of $C^k$ differentiable structures on $\mathbb R$.

There are infinitely many differentiable structures on $\mathbb R$ : take any homeomorphism which is no diffeomorphism (such as $x\mapsto x^3$), and you get an non-usual differentiable structure on $\mathbb R$!

Even better : there exist uncountably many different real analytic structures on $\mathbb R$. But this example is general : for every $k\in \mathbb N \cup \{\infty,\omega\}$, the group Homeo($\mathbb R$) acts transitively on the set of $C^k$ differentiable structures on $\mathbb R$.

EDIT : If you want to show that up to homeomorphism, there exists only one differentiable structure on $\mathbb R$, just remember that (more generally) all $1$-dimensional non-compact manifolds are diffeomorphic to $\mathbb R$. [this is proved in Lafontaine e.g]