In Goldstein's Classical Mechanics, in the chapter on Rigid Body Mechanics he establishes the fact that no matter what point you choose on the Rigid Body, the angular velocity remains the same. The argument is based on the fact that the angular velocity field in a rigid body must be continuous. My questions are,
1)What does continuity here mean?
2)Are there physical scenarios where the continuity of angular velocity field is violated?
3)If there are such discontinuities if at all, then to what degree do they affect the motion of the body in a qualitative sense?