- <Adj> function extension is an extension that keeps (or gives) Adj property
- extended function is usually treated as originals if extension is good enough. Real analysis starts with defining real numbers and extending familiar functions onto them
- in this particular case we do not need C - even continuous extension on R works and agrees with x/x = 1 at 0
- holomorphic (analytic) extension makes function infinitely differentiable at every point of C
- because of the nature of discontinuity you can’t extend the simple arccosh in any reasonable way on C without introducing multivalued or path-dependent functions
- this continuity makes x/x=1 a reasonable simplification for CAS imo but not for complex functions as in the OP
- many things with point singularities in R have more structure in C, but x/x is not one of them. Even 1/x is of a different nature.
“You do not divide by zero” that forces you to carry x != 0 is more of a high-school construct than a real thing. Physicists ignore even more important stuff, and in the end their formulas work “just fine”.
It was really fun
This is completely different from the phenomenon described in the article: arccosh discontinuity can’t be dealt the same way. In fact complex analysis prefers to deal with it my making functions path-dependent (multi-valued).