The proof I'm referring to is this arxiv paper
By the continuity of F , there exists an open non-degenerate n-dimensional parallelepiped X, centered at $a$ and contained in X ′ , whose edges are parallel to the coordinate axes such that for every x in X we have $F (x, b_1 ) < 0$ and $F (x, b_2 ) > 0$
I can see how if $x < a$ and $x>a$ it $F$ would be $<0$ and $>0$ but I can't see why the $b$ variable would determine if the function is greater or less than zero, independently of our $x$ point. The "center at $a$" part is tripping me up. I can't visualize why this must be true.
The second question I have is of this section:
let $t \ne 0$ be small enough so that $x + te_j$ belongs to X. Putting $P = (x, f (x))$ and $Q = (x+te_j , f (x+te j )) $, we notice that F vanishes at P and Q.
Why do we notice Q vanishes? Is it because we've chosen $t$ to be small enough that we can say that $f(x)$ is linear?