CuriouslyNuclear

• CuriouslyNuclearJun 27, 2025, 8:57 PM

  1 point

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  on: De­fault his­tory is dead wrong

  This post ends before it even starts bothering to make arguments for why its central claim is true. It just repeats the claim with increasingly angry and profane language.

• CuriouslyNuclearJun 13, 2025, 2:36 AM

  5 points

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  on: Dis­con­tin­u­ous Lin­ear Func­tions?!

  This is a good post. I remember being so confused in a real analysis class when my professor started talking about how important it is that we restrict our attention to continuous linear functions (what on Earth was a discontinuous linear function supposed to be?). This post explains what’s going on better than my professor or textbook did.

  I agree with one of the other commenters that this part is not technically phrased accurately:

  One way to think about continuity is that a function not having any “jumps” implies that it can’t have an “infinitely steep” slope

  Because eg the derivative of $f\left(x\right)=\sqrt[3]{3x}$ at $x=0$ is $+\infty$ despite the fact that it’s continuous there.

• CuriouslyNuclearJun 13, 2025, 2:29 AM

  6 points

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  in reply to: Jeffrey Liang’s comment on: Dis­con­tin­u­ous Lin­ear Func­tions?!

  For your example to work you need to restrict the domain of your functions to some compact e.g. $C^1\left(\right[0,1\left]\right)$because the uniform norm requires the functions to be bounded.

  Hmm, is this really a substantive problem? Call it an “extended norm” instead of a norm and everything in the post works, right? My reasoning: An extended norm yields an extended metric space, which still generates a topology — it’s just that points which are infinitely far apart are in different connected components. Since you get a perfectly valid topology, it makes perfect sense for the post to talk about continuity. Or at least I think so; am I missing something?