Is this a $c^\infty$ Atlas?

Question

I'm watching Frederic Schuller's "Lectures on the Geometric Anatomy of Theoretical Physics," and came across the following in this video: We want to define an atlas on the real line with the standard topology. We choose the atlas to be $\mathscr{A} = \{ (\mathbb{R},x) \}$ where $x(a)=\sqrt[3] a.$

Schuller states that this is a $C^{\infty}$ atlas. I'm confused about how this can be since $\sqrt[3] a$ is not differentiable at 0. Am I misunderstanding definitions here or has Schuller made a mistake?

Answer 1

You want the transition functions to be smooth, you don't give a damn about the charts themselves.