So much

Comments

• jeena@piefed.jeena.net ⁨8⁩ ⁨months⁩ ago

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

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• weker01@sh.itjust.works ⁨8⁩ ⁨months⁩ ago

  Let epsilon < 0.

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• davidagain@lemmy.world ⁨8⁩ ⁨months⁩ ago

  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.

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  • affiliate@lemmy.world ⁨8⁩ ⁨months⁩ ago

    unless f(x~0~ ± δ) is some kind of funky shorthand for the set { f(x) : x ∈ ℝ, | x - x~0~ | < 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.

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    • davidagain@lemmy.world ⁨8⁩ ⁨months⁩ ago

      There’s notation for that - (x0 - δ, x0 + δ), so you could say
      f(x0 - δ, x0 + δ) ⊂ (L - ε, L + ε)

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• Jerkface@lemmy.world ⁨8⁩ ⁨months⁩ ago

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

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• hexaflexagonbear@hexbear.net ⁨8⁩ ⁨months⁩ ago

  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.

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  • Zagorath@aussie.zone ⁨8⁩ ⁨months⁩ ago

    x_0 - delta and x_0

    Lemmy actually supports proper subscript (though not not clients do). Surround with tildes.

    x~0~ - δ is x~0~ - δ

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• swab148@lemm.ee ⁨8⁩ ⁨months⁩ ago

  Yeah

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• affiliate@lemmy.world ⁨8⁩ ⁨months⁩ ago

  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

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  • someacnt_@lemmy.world ⁨8⁩ ⁨months⁩ ago

    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.

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    • affiliate@lemmy.world ⁨8⁩ ⁨months⁩ ago

      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

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• model_tar_gz@lemmy.world ⁨8⁩ ⁨months⁩ ago

  Calculus, Motherfucker! Do you speak it?!

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• 10_0@lemmy.ml ⁨8⁩ ⁨months⁩ ago

  I’ll invest

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• emergencyfood@sh.itjust.works ⁨8⁩ ⁨months⁩ ago

  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.

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  • davidagain@lemmy.world ⁨8⁩ ⁨months⁩ ago

    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.

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