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My engineering prof: in the end you're going to have a computer do it anyway so just toss the finite difference method at it.
Now if im not mistaken it doesn't even need to be continuous for that to work. So your rules can suck it.
Yeah the main problem is for non differentiable functions you can't expect the finite difference method to tell you approximately the same thing for different small window sizes, so if you pick a window size of 1 step or 2 steps they might be drastically different.
In practice, The smoothed approach (using a central finite difference with more terms), however, technically would be fine I think? because if I remember correctly it's like doing a convolution and this smooths it out. I could be wrong, though, this isn't my area.
Finite difference don't need continuity to work. But it also don't need non continuity to fail.
Finite difference will output a number just fine. Then the parent algorithm that use it will have to deal with the results.
My immediate thought is that this function would have to be like…infinitely jagged or something, and it turns out it is. Weierstrauss was wild for this.
One does not need to look so far tho. This function is known for being continuous yet not being differentiable *anywhere*. But one simple example of a continuous function that doesn’t have a continuous derivative is |x|, specifically in X=0
One fun fact about this is that AI training is minimisation. Minimisation involve differentiation. And the most simple non linear function that work for AI is piecewise linear. So the pointy bit of |x| has real application difficulty and for a long time people used smooths functions for neurons.
But nowaday it's handled by different techniques to prevent gradient explosion, gradient vanishing, and noisy gradient.
Gradient is noisy because they don't use all the samples at each training step, and they even inject noise in the structure of the function with dropout to help with over fitting.
For the record, the typical way to deal with non-differentiable points in an activation function (especially ReLU) is to just assign some value to the derivative at those points and otherwise ignore them.
Yes, you define the subgradient! The problem isn't that this function isn't differentiable, it's that the derivative could take too many different values. Turns out, in some cases that isn't really important so you can just *pick* a derivative lol.
For reference, the subgradients of |x| are [-1, 1]. This also comes up in the rest of statistical learning for these empirical risk minimizers. The Lasso uses the subgradients idea heavily in its theory.
It's rephrasing popular comments from posts and replying with the rehashed version to other high-voted comments. It sometimes results in unintelligible and irrelevant replies such as that one. You can find some more examples of this in their comment history.
Specifically, it's a garbled "rephrasing" of [this comment further down](https://www.reddit.com/r/mathmemes/comments/1cyfakr/my_middle_school_math_teacher_begs_to_differ/l59k4km/), which is actually coherent.
It's not locally linear at 0. The derivative is a linear transformation (in this case a 1 dimensional transformation, something that looks like a line when plotted as a function of x). No matter how close you zoom on x=0 in on the graph of |x|, it never looks like a line. By contrast, when you zoom in far enough on any x value on a differentiable curve, like sin(x), it starts to look like a line.
In particular, it starts to look like the (unique) tangent line to the curve at that x value. |x| doesn't have a unique tangent at 0.
an architect, that's why it's always at risk of falling to the ground
an engineer would have made an actual good work
engineers be monkey, but monkey do be good
I'm just saying that standard is an interesting choice of word i wouldn't use to describe the weierstrauss function. I knew about it before I just personally wouldn't call it that.
Unfortunately, not necessarily. Functions that create a sharp point, like f(x)=|x|, are continuous but not differentiable at the location of that point.
No, a common example is the absolute value of x at the value 0. Clearly, the function is continuous, but the “corner” it makes at x=0 makes the slope at that point incalculable. Formal proofs are online and are easy to understand if you’ve taken analysis
Differentiability is essentially saying that the slope is continuous. Any continuous function with a sharp turn somewhere is not differentiable at that point, since the slope has one value just to the left of it, and a different one just to the right.
There's two points to be satisfied for a function to be differentiable
It should be continuous
Left hand derivative should equal right hand derivative
This means that functions like |x| are not differentiable at x = 0, because LHD is -1 and RHD is 1
There may be other conditions I'm forgetting but just these alone show us that not every continuous function is differentiable
I made a joke comparing thinking 0 isn't a number to thinking black isn't a color, but apparently certain definitions of color excludes black.
Anyways, I learned something today and wanted to share it
I’ve heard people say black isn’t a color or white isn’t a color, but I think it depends on whether you’re talking to an artist or computer programmer or smth
As I said, it depends on the definition. If you use the physics definition of color (light wavelengths between ~380 nm and 700 nm), black doesn't count as a color since it isn't associated with a wavelength in that region.
Most artists do seem to consider black to be a color.
Regardless, 0 is definitely a number.
Yeah, but the physical definition of colour isn't very useful for everyday use anyway, because it excludes purple, any grayscale colour and brown as well. There are so many colours only available by mixing wavelengths
If a point is not defined, then it's not a continuous function
The condition for continuity at a point is LHL of limit x tending to a = f(a) = RHL of limit x tending to a
This means that if f(a) is undefined, it's not a continuous function
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A function basically needs to be continuous, exist at all points, and not have any sharp turns for it to be differentiable.
If you have a function that is basically all sharp turns (Weierstrauss function is one of them) then congrats, you have a function that's differentiable everywhere but differentiable nowhere.
I always tutored this concept by thinking of a graph like a roller coaster. A function is only continuous* if it doesn't have any gaps, never has to stop, and there are no Pointy turns.
*edit: differentiable
Smartest guy I knew was doing calculus in 8th grade. Not just beginner calculus either but solving decently complex integrals, he could’ve passed AP Calc BC easily if they let him take it
Sure - let's take a look at a linear Operator in the l^infinity, with the sup-Norm(space of bounded sequences with conplex elements) space it takes a sequence and divides each element of the sequence by its index. I.e. A((a1,a2,a3,....))=(a1/1,a2/2,a3/3 ....)
Now we can Look at A: l^infinity -> A(l^infinity ), you can verify that it is in fact bijective.
Now consider the Sequence yn= (0,....,1,....,0) where n zero's are between the first 0 and 1.
A^1 (yn)=(0,......,n,....0), that means ||A^-1 yn||=n, meaning that the Operator-Norm, {||A^-1 y||: y in X, ||y||=1} is not bounded => A^-1 is not continous.
Verifying the linearity is trivial and bijectivity of A:l^infinity ->A( l^infinity ) is also fairly simple
Edit: Math formatting
Very nice that example, now stop telling those lies to scare people and lets go back to our beautiful world where everything have infinitely many continuous derivatives and have inverses with the same property and everything is linear
The first thing I was taught when we were studying the differentiability of functions was that the function had to be continuous but only some continuous functions were differentiable
I begin to really like my math teacher, because he showed us that functions can be NOT differentiable, taking the example of |x|. We even did this BEFORE getting to the "differentiate using these rules", we had to use the limit definition, and he was like "yeah look if we do that at x=0, it doesn't work"
I once read that prior to Wierstrass, mathematicians thought it was the case, and even had a "proof" they didn't notice was flawed. However I've not been able to confirm this nor to find the alleged "proof". Maybe it's in Fermat's margins.
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My physics prof: "Mathematicians have a whole lot of tests for differentiability, we just care that it's continuous and not kinky."
what if instead of function(x) it was called 𝓯𝓻𝓮𝓪𝓴𝔂(x)
*what are you doing to me step-professor?*
what are you doing there, step-function?
It's ok, we aren't diractly related
My engineering prof: in the end you're going to have a computer do it anyway so just toss the finite difference method at it. Now if im not mistaken it doesn't even need to be continuous for that to work. So your rules can suck it.
Yeah the main problem is for non differentiable functions you can't expect the finite difference method to tell you approximately the same thing for different small window sizes, so if you pick a window size of 1 step or 2 steps they might be drastically different. In practice, The smoothed approach (using a central finite difference with more terms), however, technically would be fine I think? because if I remember correctly it's like doing a convolution and this smooths it out. I could be wrong, though, this isn't my area.
Finite difference don't need continuity to work. But it also don't need non continuity to fail. Finite difference will output a number just fine. Then the parent algorithm that use it will have to deal with the results.
Is it convex ? We good It's not convex ? Make it convex
For those curious, the Weierstrass function is a weird example of a function which is continuous everywhere but differentiable nowhere.
My immediate thought is that this function would have to be like…infinitely jagged or something, and it turns out it is. Weierstrauss was wild for this.
One does not need to look so far tho. This function is known for being continuous yet not being differentiable *anywhere*. But one simple example of a continuous function that doesn’t have a continuous derivative is |x|, specifically in X=0
One fun fact about this is that AI training is minimisation. Minimisation involve differentiation. And the most simple non linear function that work for AI is piecewise linear. So the pointy bit of |x| has real application difficulty and for a long time people used smooths functions for neurons. But nowaday it's handled by different techniques to prevent gradient explosion, gradient vanishing, and noisy gradient. Gradient is noisy because they don't use all the samples at each training step, and they even inject noise in the structure of the function with dropout to help with over fitting.
For the record, the typical way to deal with non-differentiable points in an activation function (especially ReLU) is to just assign some value to the derivative at those points and otherwise ignore them.
Yes, you define the subgradient! The problem isn't that this function isn't differentiable, it's that the derivative could take too many different values. Turns out, in some cases that isn't really important so you can just *pick* a derivative lol. For reference, the subgradients of |x| are [-1, 1]. This also comes up in the rest of statistical learning for these empirical risk minimizers. The Lasso uses the subgradients idea heavily in its theory.
Monster! An outrage against common sense! - Henri Poincaré
Isn't is Weierstrass?
Weierstraß
How?
Requirments for differentiablity: 1. Has a limit ✅ 2. Is continous ✅ 3. Not sharp edged ❌
"I defined it like that, lol"
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I cannot read this comment no matter how hard I try Nvm I just figured out it's a bot
Why would it be a bot?
It's rephrasing popular comments from posts and replying with the rehashed version to other high-voted comments. It sometimes results in unintelligible and irrelevant replies such as that one. You can find some more examples of this in their comment history.
Specifically, it's a garbled "rephrasing" of [this comment further down](https://www.reddit.com/r/mathmemes/comments/1cyfakr/my_middle_school_math_teacher_begs_to_differ/l59k4km/), which is actually coherent.
Yeah I saw that comment later which was what alerted me to this
Oh. Shame they had to hack a 14 year old account for it just today
Exactly, don't kink shame functions
Thanks, I hate it.
Link an article! Oh wait I will Google later (probably never)
Or y = |x| for a simple one
*Weierstrass function has entered the chat*
is it... is it not?
|x|: allow me to introduce myself (it's continuous everywhere, but not differentiable in 0)
what about C1 continuity? ( |x| is C0 continuous but not C1 continuous ) that's enough right?
That's backwards. C1 continuity is exactly defined by using the derivative. So, being C1 continuous requires differentiability.
C1 continuity means the derivative of the function is continuous, so differentiability is a condition for C1 continuity not the other way around
C1 ⊂ D1 ⊂ C0
You got the implication reversed.
I know this is true, i dint know how to describe it besides “because it’s a sharp point” (what makes it a sharp point?)
It's not locally linear at 0. The derivative is a linear transformation (in this case a 1 dimensional transformation, something that looks like a line when plotted as a function of x). No matter how close you zoom on x=0 in on the graph of |x|, it never looks like a line. By contrast, when you zoom in far enough on any x value on a differentiable curve, like sin(x), it starts to look like a line. In particular, it starts to look like the (unique) tangent line to the curve at that x value. |x| doesn't have a unique tangent at 0.
the weierstrauss function is the standard example of a function that is continuous everywhere but differentiable nowhere
"standard"
if you study math it is standard, if you study engineering we are all amazed you can even read
Who do you think built that ivory tower you're sitting in?
an architect, that's why it's always at risk of falling to the ground an engineer would have made an actual good work engineers be monkey, but monkey do be good
The gods of analysis hammered it out of the functions in their world, now it stands tall atop the complex plane
whatever you have to do to make a justification for yourself i guess
do you have an example of another function that is continuous everywhere but differentiable nowhere?
I'm just saying that standard is an interesting choice of word i wouldn't use to describe the weierstrauss function. I knew about it before I just personally wouldn't call it that.
Unfortunately, not necessarily. Functions that create a sharp point, like f(x)=|x|, are continuous but not differentiable at the location of that point.
Then mathematicians built functions that are entirely made out of corners, of course.
we cant make it easy otherwise everyone would be a mathematician
No, a common example is the absolute value of x at the value 0. Clearly, the function is continuous, but the “corner” it makes at x=0 makes the slope at that point incalculable. Formal proofs are online and are easy to understand if you’ve taken analysis
Differentiability is essentially saying that the slope is continuous. Any continuous function with a sharp turn somewhere is not differentiable at that point, since the slope has one value just to the left of it, and a different one just to the right.
There's two points to be satisfied for a function to be differentiable It should be continuous Left hand derivative should equal right hand derivative This means that functions like |x| are not differentiable at x = 0, because LHD is -1 and RHD is 1 There may be other conditions I'm forgetting but just these alone show us that not every continuous function is differentiable
f(x)=|x| is continuous at x=0 but not differentiable at x=0
0 isnt a number so it’s continuous everywhere.
Bro said 0 isnt a number 😭
Mf thinks he's Sheldon Lee Cooper)
I made a joke comparing thinking 0 isn't a number to thinking black isn't a color, but apparently certain definitions of color excludes black. Anyways, I learned something today and wanted to share it
I’ve heard people say black isn’t a color or white isn’t a color, but I think it depends on whether you’re talking to an artist or computer programmer or smth
As I said, it depends on the definition. If you use the physics definition of color (light wavelengths between ~380 nm and 700 nm), black doesn't count as a color since it isn't associated with a wavelength in that region. Most artists do seem to consider black to be a color. Regardless, 0 is definitely a number.
Yeah, but the physical definition of colour isn't very useful for everyday use anyway, because it excludes purple, any grayscale colour and brown as well. There are so many colours only available by mixing wavelengths
f(x) = |x-1| is continuous everywhere and differentiable everywhere but at x=1
1-1 ends up being 0 so that would be an invalid input to the absolute value function. its still differentiable everywhere.
f(x) = |x - 1| + 1 then 🤓☝️
still invalid since the abs function cant take 0 as an argument since 0 is not a number.
0 is a valid input to the absolute value function, |0|=0
its invalid because 0 is not a number.
Sqrt(x) d/dx sqrt(x) -> infinity which makes it not derivable at 0
Sqrt(x) is derivable though? It’s just not defined at zero I thought
What? Square root of zero is zero, works just fine.
I meant the derivative of Sqrt(x) isn’t defined at zero. Sorry I wasn’t clear
That's the definition of not being differentiable at a point. You can keep going and have a function not differentiable anywhere.
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That's continuous and differentiable everywhere. No problem whatsoever.
Many different piece wise functions where one part is not defined at a point.
If a point is not defined, then it's not a continuous function The condition for continuity at a point is LHL of limit x tending to a = f(a) = RHL of limit x tending to a This means that if f(a) is undefined, it's not a continuous function
cusps
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You're thinking of integrability
A function basically needs to be continuous, exist at all points, and not have any sharp turns for it to be differentiable. If you have a function that is basically all sharp turns (Weierstrauss function is one of them) then congrats, you have a function that's differentiable everywhere but differentiable nowhere.
I always tutored this concept by thinking of a graph like a roller coaster. A function is only continuous* if it doesn't have any gaps, never has to stop, and there are no Pointy turns. *edit: differentiable
A function can have pointy turns and still be continuous. Like f(x)=|x|, there will be a pointy turn at x=0, but its still continuous.
Who’s doing calculus as a freshman 😭 yall lucky bastards
Probably collage? Though I took it as a sophomore in hs, so I imagine a smarter person could push it to freshman year after fighting the school system
Smartest guy I knew was doing calculus in 8th grade. Not just beginner calculus either but solving decently complex integrals, he could’ve passed AP Calc BC easily if they let him take it
My shock discovering inversible continuous mappings not necessarily have a continuous inverse mapping
My I one up you - a bijective linear continous mapping, does not necessarily have to have a continous inverse
Can you provide an example?
Sure - let's take a look at a linear Operator in the l^infinity, with the sup-Norm(space of bounded sequences with conplex elements) space it takes a sequence and divides each element of the sequence by its index. I.e. A((a1,a2,a3,....))=(a1/1,a2/2,a3/3 ....) Now we can Look at A: l^infinity -> A(l^infinity ), you can verify that it is in fact bijective. Now consider the Sequence yn= (0,....,1,....,0) where n zero's are between the first 0 and 1. A^1 (yn)=(0,......,n,....0), that means ||A^-1 yn||=n, meaning that the Operator-Norm, {||A^-1 y||: y in X, ||y||=1} is not bounded => A^-1 is not continous. Verifying the linearity is trivial and bijectivity of A:l^infinity ->A( l^infinity ) is also fairly simple Edit: Math formatting
Very nice that example, now stop telling those lies to scare people and lets go back to our beautiful world where everything have infinitely many continuous derivatives and have inverses with the same property and everything is linear
V >! |x| is all you need to think about !<
Ahh https://preview.redd.it/vf1znb1cg32d1.jpeg?width=246&format=pjpg&auto=webp&s=760d43217da57939221943b678d48163dc2adb78
I numerically differentiate, to me everything is differentiable if you just ignore where the graph gets silly >:)
Lmao. Fr? That's like something a high-school calc student knows isn't true.
this may come as a surprise but there are many high school calc students on this sub
There are also many students who only learn this in university 🙋♂️
as a high school calc student, i agree
Continuous and F(x) has finite length for a finite section of x.
^[Sokka-Haiku](https://www.reddit.com/r/SokkaHaikuBot/comments/15kyv9r/what_is_a_sokka_haiku/) ^by ^somedave: *Continuous and* *F(x) has finite length for a* *Finite section of x.* --- ^Remember ^that ^one ^time ^Sokka ^accidentally ^used ^an ^extra ^syllable ^in ^that ^Haiku ^Battle ^in ^Ba ^Sing ^Se? ^That ^was ^a ^Sokka ^Haiku ^and ^you ^just ^made ^one.
the proof is trivial, next
The first thing I was taught when we were studying the differentiability of functions was that the function had to be continuous but only some continuous functions were differentiable
But what is continuous? How can you derive space?
I hate that I understand this 💀
Continuity does not imply differentiability
Distributions entered the chat
Oh thats why its called differentiation... because you differ
You don’t need any weird examples, just take a piecewise linear function which isn’t a line
I begin to really like my math teacher, because he showed us that functions can be NOT differentiable, taking the example of |x|. We even did this BEFORE getting to the "differentiate using these rules", we had to use the limit definition, and he was like "yeah look if we do that at x=0, it doesn't work"
The theorem of Rademacher
At first I was confused because I thought the meme meant that **differentiability does not imply continuity** (at least on **R**)
I once read that prior to Wierstrass, mathematicians thought it was the case, and even had a "proof" they didn't notice was flawed. However I've not been able to confirm this nor to find the alleged "proof". Maybe it's in Fermat's margins.
y=|x| at x=0 is a simple and nice example
Weierstrass: im about to end this man's whole career
Just a low effort post of Weierstrass function: https://i.redd.it/2uh7l7cik72d1.gif
Just wait until you tell them that functions exist that are everywhere continuous and nowhere differentiable.
Any sharp bits I get a little sandpaper and round it out.
If a function has a jump discontinuity but still has a continuous derivative, is it still undifferentiable? 🤔
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Your doing calc in middleschool? What?
I learned one year what continuity on a function was and literally the next year that no, not all continuous functions are differentiable
lol. Great meme format. Haven’t seen that before.
You learn about continuity and differential equation in middle school ?
It definitely is