Feels a bit repetitive at moments, in particular the “objections” section didn’t feel useful to me once they clearly stated what the usable definitions and properties were, but it does not seem that useless considering that they kept getting similar objections.

I have a comment about the limit argument, in particular I disagree with the statement :

The issue is that the counterargument assumes that if the limit exists and f(0) is defined, then lim_(x -> 0) f(x) = f(0). This isn’t always true: take a continuous function and add a point discontinuity.

I would say that it is true that if the limit exists and f(0) is defined, then lim_(x -> 0) f(x) = f(0); as the definition of limit includes ultimately constant sequences of domain points. What happens when you add a discontinuity point is that the limit will not exist anymore because the limit through ultimately constant sequences of values will be different from the limit by different values. What they call “limit” is what I call “limit by different values”.

And just when I finished typing this paragraph, I just read on the French Wikipedia that they use a deprecated definition of limit while Belgian school and university tought me the modern one. English Wikipedia still gives the old one though (through 0<|x-x0|<delta).
Here is the relevant French comment :

Cette définition moderne, cohérente avec la définition topologique générale (voir infra) et désormais en vigueur en France, supplante la définition historique de Weierstrass, appelée aussi « limite épointée » ou « limite par valeurs différentes », enseignée encore parfois dans les universités françaises et dans d’autres pays

translating as

This modern definition, consistent with the general topological definition (see below) and now in use in France, supplants the historical definition of Weierstrass, also called “blunt limit” or “limit by different values”, still sometimes taught in French universities and In other countries

Vocabulary dispute aside, I do agree with the core argument that the newly defined operation does not need to be continuous, and in fact the domain of x -> 1/x on the reals cannot be continously extended to 0 as the limit by inferior values is -infty while the limit by superior values is +infty.