I'm reading the the following theorem from C.H. Edward's Advanced Calculus of Several Variables: Theorem 1.4, pp. 167
How are the first two conditions derived. I know that the intermediate value theorem is used, but I'm not sure how the sign of the functions is pinned down. For example, why can't it be the case that the signs are the same; that is, why can it be the case that $G(x, d_1) > 0$ and $G(x, d_2) > 0$. Also, if the rectangle chosen is such that the graph crosses the horizontal lines, then can't we have the case that $G(x, d_1) = 0$ for some $x \in [c_1, c_2]$ or $G(x, d_2) = 0$ for some $x \in [c_1, c_2]$.
What assumptions are imposed to get to these specific conditions.
The rest of the motivation for the proof is clear.