• NocturnalMorning@lemmy.world
    7·
    6 months ago

    How is a function continuous everywhere but not differentiable anywhere? Thats like an oxymoron almost. Is every point a step function, a corner? Im so confused…so many questions.

    • m_‮fOPAEnglish
      22·
      6 months ago

      Fractals, baby!:

      In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is also an example of a fractal curve.

      The Weierstrass function has historically served the role of a pathological function, being the first published example (1872) specifically concocted to challenge the notion that every continuous function is differentiable except on a set of isolated points. Weierstrass’s demonstration that continuity did not imply almost-everywhere differentiability upended mathematics, overturning several proofs that relied on geometric intuition and vague definitions of smoothness.

      Animation based on the increasing of the b value from 0.1 to 5. ☝️

      • NocturnalMorning@lemmy.world
        11·
        6 months ago

        I dislike this very much! My engineer brain never studied fractals, and was told that functions that are continuous everywhere will be differentiable…this is so counterintuitive! I find it funny that the first fractals were literally introduced to show that this assumption was not universally true. Basically, the mathematical equivalence of trolling 🤣

    • SethranKada@lemmy.caEnglish
      9·
      6 months ago

      Its a type of fractal, I think. A pretty famous one too, though I don’t remember anything about it besides that it exists.