A multivalued function is not a function. Hence the need to call it something else. In complex analysis you have branch cuts and things like that. We don’t call something a function that doesn’t have a unique output for a given input.

EDIT: Complex analysis was developed well before the formal definition of functions, relations, set theory. As such there are legacy terms in use. Mathematics has a lot of abuse of notation. For instance x=2 can mean assignment or an equation to solve. It depends on context.

In each branch of a multi-function the mapping is a function and, as a convenience, a single term “multi-function” is used to encapsulate all this. It cuts out the verbosity. A multi-function is not a function though.

My pedantry in this comes from the blog post. It was poorly written and left out some, from a mathematical point of view, important details. For instance the author uses a Schauder basis and has an uncountable sum at one point whilst ignoring the need distinguish between a Hamel basis and Schauder basis or the need to use a linear ordering in the reals to properly make sense of the uncountable sum.