In the document That Sound the Same by S. J. Chapman page $127$, he explains the following concept :
In order for the first derivative of the tranposition to be continuous it is sufficient that
($1$) Every fold lies along an outside edge of the new shape.
($2$) Each edge of each copy of the original shape that lies in the interior of the final shape must be adjacent to its reflection on an associated copy of the original shape.
After that fact, he cited that "For example, the first derivative of the transposition is discontinuous across the lines ($1$) and ($2$) in Figure $4$". I think because the English language is not my first language, I'm not able to well understand that line. Could anyone be able to explain in details why it is true?
Clarification : The method of transposition has also recently been used by Buser to generate new examples of isospectral plane domains. We could find an explanation of what is a transposition (or transplantation) in Isospectral Riemann surfaces by Peter Buser.
Please, if there are some corrections to inform me.
Thanks in advance!