i guess basically i think there's two meanings of function, one is like "the thing we do calculus on", and the other is a synonym for a mapping. And historically, how I think it went is that the graph of a function is a mapping, and for a while people didn't really think hard about this stuff. And maybe the weierstrass function is an important moment—showing that a function can't quite be "thing you do calculus to". So then functions were formalized as mappings and ended up as synonymous, and now you can use "function" to refer to mappings even when you can't do calculus in that setting. But the preformal calculus usage survived. And it's something like "number", a vague term that doesn't pick out a particular formal structure (such as rational vs reals vs hyperreals), but associated to calculus the way number is associated to arithmetic