The problem says
On $\mathbb{R}^1$consider $f(x)=x$ and $g(x)=x^{1/3}$ both $\mathbb{R} \to \mathbb{R}$. Consider atlases $\alpha_1=\{(\mathbb{R},f)\}$ and $\alpha_2=\{(\mathbb{R},g)\}$. Show that both atlases define a $C^{\infty}$ structure on $\mathbb{R}$ (I've done this) and explain why $\alpha_1$ and $\alpha_2$ aren't compatible.
Now, I see no reason why they aren't compatible, looking at the definition. I need that the transiition functions $f (g^{-1})$ and $g(f^{-1})$ both to be $C^{\infty}$ in this case.
However, the solution says
Consider the transition function $f(g^{-1})$. This satisfies $f(g^{-1 })(x)=x^{1/3}$ which is not differentiable hence not smooth.
Uhm, it is...? I mean, here $(x^{1/3})'=\frac{1}{3}x^{-2/3}$ there's a derivative. yep it's differentiable. And I can keep on going. Is "differentiable" defined in different ways in mathematics depending on the area??? This is "differentiable manifolds" and I don't recall it being different.
why is $x^{1/3}$ not differentiable?