When checking the transition maps for differentiability in order to determine if a manifold is differentiable, do we fill in any removable singularities (i.e. simplify the function composition before differentiating)? Or do these removable singularities make a difference?
The example I'm thinking about is the manifold $\mathbb{R}$ with the single chart $x\mapsto\sqrt[3]{x}$.
If you don't ignore removable singularities then this atlas is only $C^0$ since it's neither compatible with itself nor (by the chain rule) any other atlas beyond $C^0$.
But if you do ignore removable singularities this leads to odd behavior, such as a smooth manifold with a chart that isn't even differentiable.