Denjoy's Counterexample, Semi-Conjugation, Irrational Rotation, $C^1$ Diffeomorphism and Cantor Set in $\mathbb{R}$

Question

Introduction to the Geometry of Foliations, Part A. Gilbert Hector, Ulrich Hirsch. Pages 73, 74.

1. Why is the map $J$ everywhere continuous from the right and continuous from the left for points not belonging to $G(u_0)$?
2. What do they exactly mean by "the jump of $J$ in $t$ equals $l_n$"?
3. How to prove that $C=\overline{J(\mathbb{R})}$ is actually a Cantor Set in $\mathbb{R}$?