A confusion about find the boundary of a set $E = D \times E_x$, where $D \subseteq \mathbb{R}^n $ and $E \subseteq \mathbb{R}^1 $

Question

In the book of Mathematical Analysis II by Zorich, at page 132, it is given that

However, in the proof of the Remark, I couldn't understand how does the author conclude that $\partial E$ is the graphs of $\phi_i$ and $Z$, which is a portion of the product of the boundary $\partial D$ and a sufficiently large one-dimensional closed interval of length $l$ ?

Edit:

Any help or hint is appreciated.

Answer 1

In general, $\partial(A×B)=((\partial A)×B)∪(A×(\partial B))$ (which looks a lot like the product formula for derivatives, which is no coincidence!)

I believe that the two terms the author describes are exactly these, since $B$ in this case is an interval, so $\partial B$ consists of two points (which end up corresponding to the two graphs).