I've been reading this explanation (see picture and text below) about the Schwarz-Christoffel mapping. I'm not really used to this sort of argument. My question is why are all terms constant in $(21.3)$ when $z$ is in between two points? It makes sense visually - but not sure why in terms of arguments. Thanks!
http://www.mth.kcl.ac.uk/~shaww/web_page/books/complex/Chapter21Excerpt.pdf
Let $z\in\mathbb R$, with $z\in (x_i,x_{i+1})$ (open interval, in the text: "$z$ lies strictly between two of the $x_i$"), for some $i$. The arguments of the differences $z−x_j$ are all fixed, and equal to $0$ if $z>x_j$ or $\pi$ otherwise. Now, we arrive at a "jump" in the arguments exactly when the moving $z$ crosses any of the branching points $x_i$ or $x_{i+1}$ as explained in the text. As the above holds for all $i$'s, we arrive at the statement.