Rethinking Functional Limits (

[@PeterBarnes2]

Thomae's Function is a cool variation. It's Dirichlet's function, but for rationals of the form p/q the function equals 1/q. Now you have continuity on the irrationals, and I'm pretty sure the limit exists everywhere.

[@ErsagunKuruca]

Oooh. Another function: 0 for computable numbers, 1 otherwise. It doesn't have a limit at x=0, but you cannot ever start to write an example of a series that doesn't converge to 0.

[@PeterBarnes2]

@@ErsagunKuruca Constructivists hate it! Find out the one wierd function they don't want you to compute limits for!

[@bobater]

Extremely cool use of limits! The idea of looking not just at positive and negative approaches when evaluating a limit, but different sequences of numbers, really blew my mind. Great video and great presentation!

[@angeldude101]

While the formal epsilon-delta definition only came up at the end, what was shown at only 2:00 is really the exact same thing, just with less precise language and also not necessarily being symmetric. What was shown there is a very intuitive understanding of a limit, and all epsilon-delta does is phrase that intuition in formal notation.

[@vladislavanikin3398]

I would have to disagree. These are two distinctly different notions of a limit of a function stemming from generalizing two different "approaches" of a limit of a sequence. A limit of a sequence a(n) is defined as ∀ε>0 ∃N∈ℕ n>N→|a(n)-L|

[@brendanchamberlain9388]

@@vladislavanikin3398 Would you say it is incorrect to say that, even though these two definitions are “different”, they are “equivalent”? I.e., a function has a sequential limit iff it has an epsilon-delta limit, and these two limits are equal

[@vladislavanikin3398]

@@brendanchamberlain9388 In real analysis, sure, they are completely equivalent and you actually prove it in any standard course, so it wouldn't be incorrect. But if you aren't working with real numbers, things might get tricky. In fact, you can show that there are spaces where those two definitions are different, namely, Heine's definition is weaker. That means that in some topological spaces you can have a situation where Heine's definition (or actually something close to it in spirit) applies, but Cauchy's definition (or again something like it) fails. My comment, in fact, was less about this technicality and more about different "modes of thought" behind them, so to speak. But sure, if you are concerned with real or complex analysis, then they are completely equivalent.

[@lexinwonderland5741]

lovely graphics and explanation!! I appreciate the nod to epsilon-delta while still focusing entirely on cauchy sequences -- this was intuitive and educational!! I frequently refer to Real Analysis as "Calculus for Adults" lol

[@noamyakar8554]

Love it Keith! Our souls will forever be tied through our struggles in advanced calc :)

[@alexbennie]

13:45 ish Examples of two such sequences: A: a_n = n¯¹ B: b_n =sqrt(2) × n¯¹ A is always Rational & B is always Irational.

[@Simchen]

That is a wonderful animation and introduction of the concept. However I have to point out a blunder. At around 13:00 you start constructing two sequences that each should converge to zero. But what you construct is in both cases a sequence that is monotonous and bounded but doesn't necessarily converge to zero. It is easily fixed tho. Just chose your next number to be between half of the former number and zero.

[@pimcoenders-with-a-c1725]

I think it would have been easier to pick an arbitrary rational number q, and an arbitrary irrational number a, and then have the sequence be q, q/2, q/4, q/8, ... and a, a/2, a/4, a/8, ..., since any rational multiple of a rational is again a rational, and any rational multiple of an irrational number is an irrational number.

[@Simchen]

@@pimcoenders-with-a-c1725 Yes that also works. I just wanted to keep the process close to the one presented in the video.

[@academyofuselessideas]

Great! A video on the equivalence between the sequential, epsilon-delta, and open sets definitions of continuity would be cool... It would also be cool but perhaps too advanced to see in which spaces those three definitions are different (obviously, you need a metric space for epsilon-delta definition but a topological space should suffice for the other two definitions... But in general topological spaces, sequential continuity is different than continuity... this could be an interesting way of introducing nets as a generalization of sequences)... great video!

[@chennebicken372]

I think, you actually covered the epsilon delta definition, just in your own words. You said roughly speaking "A limit existing means, that for every tolerance we can find an input from that on all the outputs lie within the tolerance" The difference just being, that you handled the specific case, when the input (n) approaches infinity. The Epsilon Delta Definition only states that "The limit that x approaches a of f(x) means, that for every tolerance ε (>0) we can find an input-range [a-δ; a+δ] (δ>0), such that all x from that range map to an output that lies within the tolerance".

[@timpani112]

There is a small mistake (or omission, however you like to view it) when you give a sequence of rational numbers that converges to zero (at ~ 13:00 in the video). While it is true that the rational numbers are dense in R, there are sequences of rational numbers constructed the way you prescribe that do not converge to zero (but to, say, -1 instead). In these cases it is much better to give a concrete rational sequence that converges to zero instead of trying to give a more "general" one, as the general approach is much easier to get wrong and all we need is one sequence. For instace, one could consider the very explicit sequence (-1,-1/2,-1/4,-1/8,...) of rational numbers, and take an equally simple and explicit sequence of irrational numbers (just multiply the rational sequence given by -sqrt(2) for instance...).

[@quantumgaming9180]

Bro this is so cool

[@MathVisualProofs]

nice work here! Enjoyed this.

[@H3XED_OwO]

Fancy seeing you here, i love your proof of the golden ratio

[@FredericoKlein]

Oh thanks for this. I was playing around one day and constructed this function by mistake (I wanted to come up with a construction for a function that would give me integers, but I failed at that). Seeing on wikipedia, the construction i came up with is slightly different (I wonder if my version is correct). In any case, what I did was consider that the cross-correlation between 2 sine waves for irrational multiplicatives of their frequencies will never align, so they should integrate (from negative to positive infinity) so something that is zero, but when they are rational, they should have a value. I mean, i don't know if this is actually correct, but in any case, thanks for showing the name of the thing I thought I came up with!

[@avyakthaachar2.718]

Very well animated and explained 👍

[@enstucky]

Hey! Found your video on the SoME voting page. My process didn't allow me to write official feedback, but I wanted to give you my thoughts anyway. The animation is well-done, and it also serves an essential role for this explanation, which would not work nearly as well if it were written down. It's clear that you made this video for a less experienced audience, and I think you did a good job at usefully meeting folks where they are. I am concerned that some of them will feel like you've essentially completed the task by the 5:00 mark with the left-right explanation, before you've actually gotten to the part that you seem most interested in. Because of this, the middle section feels a bit audience-less to me, even if the explanation is still appropriate for these more novice viewers. I'll conclude by saying I really liked the end. It's a really empathetic touch to recognize how a DNE result might not be satisfying. Moreover, while you didn't go through a full proof, I appreciate that you addressed the mixed rational-irrational case explicitly, not sweeping that tougher business under the rug.

[@pizzarickk333]

The explanation is good actually.

[@RSLT]

Very cool! Since the limit of the function exists, what is the derivative of the modified function?

[@KeithJeroskyMaths]

Great question! Keep in mind that a function "g" is differentiable at an input value "a" if the limit as x approaches a of (g(x) - g(a))/(x - a) exists, not just if the limit exists at the input value of the function itself. It turns out that the modified Dirichlet function is not differentiable at any point in its domain ("nowhere differentiable")!

[@RSLT]

@@KeithJeroskyMaths Interesting. You have indeed shown that the limit as x approaches 0 of g(x) (with a=0) exists and is zero. What am I missing here?

[@RSLT]

@@KeithJeroskyMaths By any chance, do you happen to know the ratio of rational numbers to irrational numbers?

[@KeithJeroskyMaths]

Finding the limit as x approaches 0 of g(x) (what this video shows) is not the same thing as finding the limit as x approaches 0 of (g(x) - g(0))/(x - 0).

[@RSLT]

​@@KeithJeroskyMaths By any chance, are you familiar with any other function where the limit exists and is equal to the value of the function, but the derivative doesn't exist?

[@idrankwhatt]

Dr C would be proud

[@Hi_Brien]

Really enjoyed this one😊

[@zildian1983]

Amazing!

[@josephmaloney9116]

Came here for the math, stayed for the sexy voice

[@ガアラ-h3h]

Well let’s just be very ones since the output is either 1/0 and nothing in between the limit is undefined just like the limit of any sin function is undefined sure there’s way more irrational numbers than rational ones however we still can’t tell if it’d be 1 or 0

[@BlockOfRed]

Let f : R → R be the modified Dirichlet function, i.e., f(x) = 0 if x is irrational and f(x) = x if x is rational. Let (x_n) be any real sequence such that x_n → 0. We show that f(x_n) → 0. Let ε > 0 and choose N such that |x_n| < ε for all n ≥ N. Then, for all n ≥ N, we have |f(x_n)| = max(0, |x_n|) < ε. Hence, f(x_n) → 0.

[@naturegirl1999]

I tried graphing the equation at 2:39 and didn’t get the gap at 2, what did I miss?

[@KeithJeroskyMaths]

How did you try graphing it (graphing calculator, desmos.com, etc.)? I pasted "(-x^2 + 5x - 6) / (x-2)" into desmos, and clicking on x=2 shows "(2, undefined)".

[@naturegirl1999]

@@KeithJeroskyMaths oh, I thought it would show the circle gap, I graphed it again clicked it and it did say undefined. Thanks

[@blusham4629]

Great

[@gabrielmartins7642]

Couldn't you do this to any rational point such that it never converges in any rational point ? Edit: like it seems possible to find a sequency of irrational numbers that converges to a rational one. Like a/(b+-sqrt(2)/n) a and b are integers, n goes from 1 to infinity

[@sdurien]

Teach me more things Keith

[@joansgf7515]

PapaFlammy did a video on this too.

[@stoneman172]

Usually you define functional limits either via epsilon-delta or the behavior of the function when evaluated against sequences (i.e. the idea elaborated in this video) and then prove the equivalence of both in a very (!) simple exercise as homework. I don’t see the „rethinking“ part.

[@volodymyr3169]

Looking at the title I thought it would be about functional analysis

[@degraj418]

The closest number to zero on both sides will be a rational number: -0.000...01 and 0.000...01. Is that not sufficient enough to conclude that the limit is 1?

[@KeithJeroskyMaths]

Well, remember that there is no rational number that is closest to zero on both sides...due to the density of rational numbers in the real numbers, we can always find a rational number between zero and the rational number you chose. Furthermore, due to the density of irrational numbers in the real numbers, we can always find an irrational number between zero and the rational number you chose, whose function value would be zero and not one.

[@abigailsokol3029]

the number 0.00...001 where «...» is an infinite number of zeros actually equals 0, and is not next to it. Consider «1 - 3(1/3)»

[@OMGclueless]

I don't think your method of picking sequences of rational and irrational numbers that converge to zero at 12:55 is valid. The sequence you pick might not converge to zero. For example -1-1/n is a sequence of rational numbers obeying your condition of being in between the previous element and zero and starting at a negative rational. But this sequence converges to -1 instead of 0.

[@brendanmiralles3415]

cool now integrate it

[@dicandeo9698]

Imo, the video could have ended after 4 and a half minutes, because by then you already showed that the limit to Dirichlet's function does not exist because it approaches different numbers for different input sequences.

[@vladislavanikin3398]

Good video overall, but I personally find two (or one, depends on how you count) problems with it. 1. You don't have to talk about Heine's definition of a limit in order to use "if f(aₙ) for two sequences does not converge to the same value, then the limit doesn't exist". It follows more or less directly from the standard ε-δ definition. 2. Heine's definition of a limit is not the same as this result about a limit not existing and that ideally should be at least noted. E.g. you can have the equivalence of Heine's and Cauchy's definition failing (in some topological spaces), but still have the result about a limit not existing holding. Maybe it would worth it to note this in a pinned comment. Other than that, great video, keep it up!

[@strikeemblem2886]

On 2, I suspect that he might be unaware of the subtleties beyond functions from R to R, since it seems that he just completed a first course in analysis.

[@vladislavanikin3398]

@@strikeemblem2886 Well, maybe, even though it's less of an analysis thing and more of a logic thing, IMO. I mean, those two statements are converses, so it's more of not noticing that one is A→B and the other is B→A, but it's an understandable mistake when you indeed have the equivalence of A and B, I remember doing it myself (and in this precise context as well), so it's not a dis or anything like that, just a technical note. Me talking about how in some topological spaces those two definitions may not coincide was just providing a relevant example so that it may be more clear that the two statements are not the same, but rather converses. And except this slight technicality, which doesn't really affect the overall narrative, the video is great. I just wanted to note this small bump