The mathematical explanation is given on the wikipedia page. The Picard-Lindelöf theorem states that initial value problems have unique solutions if the right h

The mathematical explanation is given on the wikipedia page. The Picard-Lindelöf theorem states that initial value problems have unique solutions if the right hand side function is Lipschitz continuous. The force created by Norton's dome is not Lipschitz continuous since it has a kink. I find it quite interesting that Newtonian mechanics only works if this condition is satisfied. Of course you cannot make a non-Lipschitz continuous force in practice, since the tip radius will be limited to the size of atoms, but theoretical physics is usually not concerned with such trivialities.