So much

Let epsilon < 0.

I feel I should understand it, but it's just outside of my reach. It's now 10 years after university.

I don't think you can use the x0 plus minus delta in the bracket (or anywhere), because then the function that's 1 on the rationals and 0 on the irrationals is continuous, because no matter what positive number epsilon is, you can pick delta=7 and x0 plus minus delta is exactly as rational as x0 is so the distance to L is zero, so under epsilon.

You have to say that
whenever |x - 0x|<delta,
|f(x) - L|<epsilon.

But I think this is one of my favourite memes.

unless f(x₀ ± δ) is some kind of funky shorthand for the set { f(x) : x ∈ ℝ, | x - x₀ | < δ }. in that case, the definition would be “correct”.
it’s much more likely that it’s a typo, but analysts have been known to cook up some pretty bizarre notation from time to time, so it’s not totally out of the question.
- There's notation for that - (x0 - δ, x0 + δ), so you could say
  f(x0 - δ, x0 + δ) ⊂ (L - ε, L + ε)

... That's enough real analysis for me today. Or ever, really.

Feel weird correcting a meme, but that should be f(x)-L where x is between x_0 - delta and x_0 + delta. As written it looks like a definition that would only work for monotone functions.

x_0 - delta and x_0
Lemmy actually supports proper subscript (though not not clients do). Surround with tildes (single tildes, rather than the double tildes of strike-through).
x₀ - δ is x~0~ - δ
Edit: and now, at least on Jerboa, the code block version is displaying incorrectly, although it now does support the actual subscript…

Yeah

i still feel like this whole ε-δ thing could have been avoided if we had just put more effort into the “infinitesimals” approach, which is a bit more intuitive anyways.

but on the other hand, you need a lot of heavy tools to make infinitesimals work in a rigorous setting, and shortcuts can be nice sometimes

Infinitesimal approach is often more convoluted when you perform various operations, like exponentials.
Instead, epsilon-delta can be encapsulated as a ball business, then later to inverse image check for topology.
- i think the ε-δ approach leads to way more cumbersome and long proofs, and it leads to a good amount of separation between the “idea being proved” and the proof itself.
  it’s especially rough when you’re chasing around multiple “limit variables” that depend on different things. i still have flashbacks to my second measure theory course where we would spend an entire two hour lecture on one theorem, chasing around ε and η throughout different parts of the proof.
  best to nip it in the bud id say

I won't ever understand advanced maths, can someone explain me?

Not an advanced mathematician, but I think it's just saying that f(x-delta) between f(x + delta) is going to give a value between L - epsilon and L + epsilon.
- I literally don't know what any of that means

Calculus, Motherfucker! Do you speak it?!

Not a mathematician, but I'm pretty sure this isn't necessarily true. What if L is -1 and f(x) = x^2? Also I think your function has to be continuous.

You're right on all three counts. It's not always true, f(x0) has to be L, and the function has to be continuous.