What is the correct graph of arccot(x)?
intmath.com
intmath.com
Oddly, that term wasn't mentioned. Instead, it feels like the author (wrongly) believes that functions must be single valued, and is trying to interpret everything within that viewpoint.
Eg, for the example value of 0.2, the answer should be more like:
acot(0.2) = -0.46365... + k π where k is an integer
https://en.wikipedia.org/wiki/Function_(mathematics):
“In mathematics, a function is a binary relation between two sets that associates each element of the first set to exactly one element of the second set”
https://mathworld.wolfram.com/Function.html:
“A function is a relation that uniquely associates members of one set with members of another set. More formally, a function from A to B is an object f such that every a in A is uniquely associated with an object f(a) in B. A function is therefore a many-to-one (or sometimes one-to-one) relation.”
Since the inverse isn’t, you have to pick, for every value in the range of f a value in its domain as the inverse value.
In general, there are zillions of ways to do that, but you want the inverse that you pick to have some desirable properties such as continuity and differentiability as much as possible. That’s why defining the inverse by making a branch cut is the way to go.
Where you make it is arbitrary, and the best choice may depend on the problem at hand (e.g. the cut that leads to the ‘nicest’ inverse may be different for arccot(x ∈ ℝ) and arccot(z ∈ ℤ). I suspect that’s why Mathematica picks an inverse that’s discontinuous at zero)
] a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. ....
] Multivalued functions of a complex variable have branch points. ... for the arctangent function, the imaginary units i and −i are branch points. Using the branch points, these functions may be redefined to be single-valued functions, by restricting the range. A suitable interval may be found through use of a branch cut ... As in the case with real functions, the restricted range may be called the principal branch of the function.
As your second link (to MathWorld) points out:
] Unfortunately, the term "function" is also used to refer to relations that map single points in the domain to possibly multiple points in the range. These "functions" are called multivalued functions (or multiple-valued functions), and arise prominently in the theory of complex functions, where the presence of multiple values engenders the use of so-called branch cuts.
I'll restate my earlier comment. For a given real x, the range of f(x)=x² is a single-element set and the range of acot(x) is a multi-element set. It does not appear the author of this intmath piece understands that the range of acot is a multi-element set, and instead believes there must be only a single "correct" primary branch.
IMO, the conclusion definitely should not be "our math definitions are not as tight as we are often led to believe" but perhaps more like "we need to remember that certain concepts are closely tied to convention and preference, and don't have a simple 'correct' answer."