im a tard
explain in tardese pls

>>16962786
Basically, it was assumed for a very long time that *continuity* implies *differentiability* in a function. There were entire proofs written out based on this presumed fact.
To translate the jargon a bit:
>continuity
Means what you probably think it means. The function is unbroken. You can zoom in on the graph as much as you want and there will be a valid y for every x.
>differentiability
This is a bit trickier if you haven't taken calculus. It basically means you can assign a rate of change to all points in that function. Imagine taking a ruler and shoving it against the curve drawn by a function. If there is exactly one angle you can hold that ruler where it can touch, but not cross through, any given point on the graph, the function is said to be differentiable.

The Weirstrauss function is a counterexample where the function is continuous everywhere snd differentiable nowhere.
It's also where the concept of fractal geometry originates.

>t. Not OP.

>>16962791
>it was assumed for a very long time that *continuity* implies *differentiability* in a function.
wtf? I thought mathematicians demand rigorous proof of everything

>>16962858
rigor is a fairly new concept compared to the full blown history of math.

>>16962858
Any logical system needs to have a set of axioms (ie. statements that are so obviously true that they can be used to prove other statements even if these statements themselves can't be proven without inventing another set of axioms). Some things necessarily have to be assumed to serve as a starting point.
Continuity = differentiability was sort of taken as one of those. The notion of "infinite jaggedness" was not really conceived of because why would it be?

>>16962761
how does this btfo anyone except miggers?

https://en.wikipedia.org/wiki/Pathological_(mathematics)

i.e its just a noisy function where it has ripples no matter how much you zoom in, yes at infinite scale too because of that infinite series so you cant define a local slope

>>16962761
I guess the derivative is:
sum[a^n b^n pi sin(b^n pi x), 0, inf]
Which for ab big enough diverges, right?

>>16962868
Despite rigor being relatively new, the rigorous definitions of continuity, differentiability, boundedness, integrability, etc. all formalize the ideas of numerical approximations.

>>16963262
true but in the context of continuity it wasn't fully explored to its limits for a long time.

>>16963197
diverges doesnt mean it doesn’t have a derivative

>>16962922
>>16962791
>>16962761
Some people often forget a crucial fact. Math exists to explain relationships between things, things that are real. These pathological functions have no validity in the physical world and so they're fiction at best. A truncated Weierstrass function is smooth and differentiable for example. Other pathological functions don't even pass the dimensional analysis test.

>>16963407
Math also exists to test observed reality against logical conclusions of continuous iteration.
Extrapolating the speed of a car beyond the path that it actually ends up taking is mathematically important and has real world implications.
Maybe the Weierstrass function doesn't describe anything "real." But the continuous-but-undifferentiable behavior it describes is common in fractal geometry. And that field describes real-world behaviors of iterative structures and their logical conclusions if they were allowed to iterate for infinite time.

>>16963430
>And that field describes real-world behaviors of iterative structures and their logical conclusions if they were allowed to iterate for infinite time.
If allowed to iterate for infinite time. That's a big if lol

>>16963439
Breakfast quandary aside, "iterate for infinite time" is useful in the real world even if the infinite iteration doesn't actually happen.
Fractal antennas are useful specifically because the mathematical structure allows for arbitrary conductor length while being confined in a finite space. So the practical engineering constraints are simplified to only need conductor width as the limiting variable.
Likewise, if we were dealing with an electrical signal that mimicked the weierstrass function at *observable* scales, that the iteration ceases below that scale becomes functionally irrelevant.

>>16963459
Being useful and being true is sometimes mutually exclusive. When you're saying some dubious concept is useful, you're surrendering to the engineers. This is exactly what happened to Ampere-Weber EM, the scientific community replaced it with Maxwellian EM for muh telegraph, antennas, etc.

>>16963485
Well the "concept" of infinity isn't dubious. The concept exists because we defined it. Whether infinities exist in material raleality isn't even relevant here.
Returning to the car analogy: a car traveling 30mph for infinite time will travel infinity miles. That is a meaningful extrapolation which says "a constant speed means you can reach any distance if given enough time." As opposed to a constantly decelerating model where, even if we removed the physical constraint that the car would have to stop eventually, there still may be a tangible limit to how far it could travel even if its motion never quite reaches zero.
This isn't just "useful." It describes real-world limitations that you'd be dealing with even if we could remove other constraints.

As far as the Ampere v. Maxwell thing is concerned, I'm not going to pretend to know the full historical context there. But Maxwell's equations do *more accurately* describe electical systems at higher frequency. This is a genuine, demonstrable, limitation of Ampere-Weber dynanics which suggests that, even if they get quasistatic systems "more correct," you suggesting that they were *generally* more correct appears to be you being arbitrary on which aspects of reality are more important to you in particular.

>>16963355
>diverges doesnt mean it doesn’t have a derivative
What are you talking about?
The derivative is infinite?

>>16963811
it still has a derivative even if it’s growing larger

>>16964065
it grows without bound for each approximation of the true function.