Quantifiers. Quantifier is one of those words which is more grandiose than the concept it represents: there are precisely two kinds of quantifiers in ordinary mathematics. (∀) means “For all” and (∃) means “There exists”. These are basic objects in logic, and are used ∀ branches of math that you can imagine [and also all the other ones].
One of the things that confuses new students about quantifiers is that the order of quantifiers is usually extremely important. When I was coming to grips with them I ended up with a model of quantified statements as a game: There are quantifiers, and then there is a condition. Going from left to right on the quantifiers, you get to choose an object when there is a ∃ and your opponent gets to choose an object when there is a ∀; all of the choices are public at the instant they are made. In the example of supercontinuity, your opponent picks the x, then you pick the delta, then your opponent picks the epsilon and the y. Now, you win if the condition is satisfied. The statement is true if (and only if) you have a winning strategy; a way to win regardless of your opponent’s choices. Explain why the strategy works, and you have a proof of the statement.
From this model a lot of general trends become more palatable: you can rearrange the order of two quantifiers if they are the same kind. If a ∃ is pulled closer to the beginning, it produces a stronger condition (because you get less information and so it is harder to win). Also, each ∃ represents some kind of construction in the proof, and you can only use a variable in that construction if it comes earlier in the chain. And you have to assume that ∀ variables are going to be chosen to make the proof as hard as possible.
This theorem arose from a tutoring exercise in which we were swapping the order of quantifiers in some multi-quantifier definitions and theorems to see what happens. Actually, if you are a tutor of a student who is struggling with proofs, this is a great exercise. There are problems like ∀/∃ x ∀/∃ y , x-y=0 that can build a lot of confidence, and then you can basically just go pull out the most quantifier-laden definitions that they’ve seen in class, work through a couple of examples and then mess around with them. It helps to do this in advance so that you can pick out good examples, though.
Longer text than usual, because I love teaching quantifiers. Every statement of mathematics feels their influence, but often people are not taught to appreciate it. This can make things very confusing. But at the same time, they’re very simple to understand, once you put some work into it. And when you do, you start to see their influence, and are a lot closer to being able to do mathematics than you were when you were blind to them.