Your first link (to Wikipedia) also has a section titled "Multi-valued functions" linking to https://en.wikipedia.org/wiki/Multivalued_function https://en.wikip

Your first link (to Wikipedia) also has a section titled "Multi-valued functions" linking to https://en.wikipedia.org/wiki/Multivalued_function https://en.wikipedia.org/wiki/Multivalued_function ] 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."