Orientability on Manifolds

Question

So I'm having some issues with the definition of orientability. I'll take the case of $\mathbb R$ for now. I know there's supposed to be only 2 orientations on a smooth, connected manifold, and for $\mathbb R$ I'm guessing these arise from the frames $x \rightarrow \frac{\partial}{\partial x}$ and $x \rightarrow -\frac{\partial}{\partial x}$. However what about the frame $x \rightarrow \frac{\partial}{\partial x}$ for $x \geq 0$ and $x \rightarrow -\frac{\partial}{\partial x}$ $x \leq 0$? Or for that matter literally any choice of $\frac{\partial}{\partial x}$ or $-\frac{\partial}{\partial x}$ for whatever $x$? These all give frames because they are clearly global sections and since the dimension is 1 they are bases for each tangent space. Then since the frame is global there is no problem with 2 sections in the frame disagreeing on their orientation as there is just 1 section and we can just define the pointwise orientation on $\mathbb R$ to be the one given by this random assignment.

Answer 1

Your assignment is not continuous around $0$, so it does not define a section. You could try to patch it by mapping $x$ to $x\frac{\partial}{\partial x}$, but then this does not yield a frame at $0$.