# Discontinuities in the angular momentum field

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?

• @Abhikumbale if it were not the same everywhere, different parts of the object would be rotating at different angular rates. Remember that angular velocity is measured in angle per time - radians per second in SI units, revolutions per minute is common in industrial settings, etc. Obviously if a rigid object is rotating $x$ times per second, all parts of it rotate at that rate. – Asher Nov 10 '17 at 4:19