I've read the paper 'Generalizations of Kochen and Specker’s theorem and the effectiveness of Gleason’s theorem', where it says that non-contextual hidden-variable theories are ruled out by a theorem of Bell (1966), which is stated as 'There does not exist a bi-valued probability function on the rays (onedimensional subspaces) of a Hilbert space of dimension greater than 2.' There is given no explanation how exactly non-contextual hidden variable theories are ruled out by it. It is clear to me that it has to do with the fact that hidden-variable theories would lead to deterministic outcomes, i.e. bi-valued probability functions. Somewhere else (unfortunately I've forgotten the source) I've read that the reasoning is that the mapping $u \rightarrow (\rho u, u)$ is continuous on the unit sphere of the Hilbert space for any density operator $\rho$, so due to Bell's 1966 theorem it cannot be deterministic.

So my first question is how to interpret the mapping $u \rightarrow (\rho u, u)$? Is $u$ a state? How does the specific observable and a specific outcome come into play in the formula? (I guess $(\rho u, u)$ is the probability for a specific outcome of a specific observable if the system is in state $u$?)

My second question is why non-contextual hidden variable theories are ruled out, but not contextual hidden variable theories?

If anything is unclear about my question, please feel free to ask. I'm looking forward to any answer or input.