You touched on the very important even more underrated concept in real analysis: limits and completeness of the real line. The discovery of limits is what, in my opinion, separates the study of mathematics into two eras: the era of vague logic based on intuition and constructs before limits and the era of rigor after the limits. Then there was the discovery of topology, which firmly grounded the concept of a limit. In my opinion, the most beautiful proof of the IVT is the topological one that uses compact sets. It works for any continuous function f: X -> R regardless of what X is.
i included the rigorous approach in the attached document :) i like to distinguish between problem-solving and rigour; the problem-solving process tends to be reverse-engineered, but the rigour needs to avoid circular reasoning
This is well said but I likely do not agree with it. Many of the mental pictures which make this comprehensible are taken from the geometry of experience. There may be physical inputs to mathematics that are hidden in a subtle way. I was immediately thinking of relativity during this presentation. There is a very similar train of thought about the order of events for places in causal contact.
Haha yes and no; there’s the theory and applications, and I guess here I’m looking at applications. I might do theory in video format (eg Dedekind cuts or Cauchy sequences approach), but we’ll see~
The problem of _f(x) = x - cos(x) = 0_ is an interesting one. Graphing the cosine function also gives some instinctive points to start with. For one thing, it must be the case that if _y = cos(x)_ then the point which satisfies _f_ (x) lies on the line _y=x._ But _y=x_ is the line of symmetry which, geometrically, reflects an invertible function into the image of its own inverse. That is the same thing as saying that if f(x) holds true then that value for x which makes it true is also where _x - arccos(x) = 0,_ which is also to say that it is the same value for _x_ which satisfies, _cos(x) = arccos(x)_ or _cos(x) - arccos(x)=0._ There is also an interesting video, by Dr. Trefor Bazett, available on YT, on this same fixed point which involves a seemingly different problem, namely evaluating nested cosine functions. The title of that gem is: *What is cos( cos( cos( cos( cos( cos( cos( cos( cos( cos( cos( cos(…?? // Banach Fixed Point Theorem*
I remember an extra credit problem from watching this. Two points on a continuous sine wave are constrained to always be L distance apart. Show that as the points travel along the sine wave (always L distance apart), the line connecting the two points at some point has a slope of zero regardless of the value of L. The key is recognizing that at some moment, the left point has a y-value of -1 and at that moment the slope of the line leading to the right has a slope >= 0. And at another moment, the left point has a y-value of +1. At that moment the slope of the line leading to the right point has a slope
cool motivation... Here is an interesting question... f continuous everywhere implies that f satisfies the intermediate value theorem (for every interval)... does a function that satisfy the intermediate value theorem (for every interval) has to be continuous everywhere?... It is also interesting that mathematicians took the intermediate value theorem for granted (meaning that they did not care about writing a proof for it since it was kind of obvious) until Bolzano in 1817
@@wabc2336 🤔 do you mean something like the constant function 1, but only defined on, let's say the intervals [0,1] and [2,3]? so it jumps from 1 to 2 horizontally? I should probably have been a bit more specific and say that a function defined over an interval, and such that it satisfies the intermediate value property in every subinterval of that interval... In any case, thanks for adding to the conversation!
"[D]oes a function that satisfy the intermediate value theorem (for every interval) has to be continuous everywhere?" No. See Conway's base-13 function, which sends every real interval to the entire real line through clever manipulation of the "decimal" representation of base 13 numbers. It is discontinuous everywhere.
I have no idea what Bring radicals are. However, it is well known that roots of quintics can be found with elliptic functions. Of course, here the question is, what do you call a "formula"?
off the top there's.... this is IVT? also MVT EVT yeah then there's FTOC but there's only 3 value theorems if i remember cuz' im doin ab calc this year
Yep! And there are implications too; am still contemplating on how to make a video on them similar to this one 😅 since there are some interesting ideas and even applications…
ManimCE! It’s the same essential program that 3Blue1Brown used; I learned the basics of Python at university year 1, then experimented with Manim throughout 2023
Very high quality video! Good job :)
Thank you very much!
I loved your video. Nice and clear explanations. Thanks Joel.
Binary search
O(log(n))
True😭
I was going to say this
Guess-n-check
Draw it on a chalkboard and measure the intercept using a ruler.
You touched on the very important even more underrated concept in real analysis: limits and completeness of the real line. The discovery of limits is what, in my opinion, separates the study of mathematics into two eras: the era of vague logic based on intuition and constructs before limits and the era of rigor after the limits.
Then there was the discovery of topology, which firmly grounded the concept of a limit.
In my opinion, the most beautiful proof of the IVT is the topological one that uses compact sets. It works for any continuous function f: X -> R regardless of what X is.
I think you meant connected sets; compact sets do help us establish the crazy useful EVT though. Totally agree with you on completeness!
@@kindiakmath Yep, connectedness of X is very important and compactness is redundant for the IVT. Good catch.
i included the rigorous approach in the attached document :) i like to distinguish between problem-solving and rigour; the problem-solving process tends to be reverse-engineered, but the rigour needs to avoid circular reasoning
This is well said but I likely do not agree with it. Many of the mental pictures which make this comprehensible are taken from the geometry of experience. There may be physical inputs to mathematics that are hidden in a subtle way. I was immediately thinking of relativity during this presentation. There is a very similar train of thought about the order of events for places in causal contact.
@@ultrametric9317 this is where rigorous definitions come in so that we won't be bamboozled by our subtle assumptions
"You won't believe" ProZD would approve.
...and 80000 people streamed it on facebook
I like the animations! Very smooth.
Thank you!
@@kindiakmath What are u using for animations?
@@freestylerveevo ManimCE!
_cos x = x_
Apparently, it's about 0.739085 (to save anyone else from getting distracted by it).
Haha I love the "about"...
@@kindiakmath
Well, strictly speaking, it is. ;)
I had a homework about this once 😂
@@m9l0m6nmelkior7 Classic exercise! XD
@@m9l0m6nmelkior7
That's bad luck, they could have asked about sin x but they gave you the hard one. ;)
The real problem here is to construct the real numbers before you can "prove" the Intermediate value theorem.
Haha yes and no; there’s the theory and applications, and I guess here I’m looking at applications. I might do theory in video format (eg Dedekind cuts or Cauchy sequences approach), but we’ll see~
The problem of _f(x) = x - cos(x) = 0_ is an interesting one. Graphing the cosine function also gives some instinctive points to start with.
For one thing, it must be the case that if _y = cos(x)_ then the point which satisfies _f_ (x) lies on the line _y=x._
But _y=x_ is the line of symmetry which, geometrically, reflects an invertible function into the image of its own inverse. That is the same thing as saying that if f(x) holds true then that value for x which makes it true is also where _x - arccos(x) = 0,_ which is also to say that it is the same value for _x_ which satisfies, _cos(x) = arccos(x)_ or _cos(x) - arccos(x)=0._
There is also an interesting video, by Dr. Trefor Bazett, available on YT, on this same fixed point which involves a seemingly different problem, namely evaluating nested cosine functions. The title of that gem is: *What is cos( cos( cos( cos( cos( cos( cos( cos( cos( cos( cos( cos(…?? // Banach Fixed Point Theorem*
I remember an extra credit problem from watching this. Two points on a continuous sine wave are constrained to always be L distance apart. Show that as the points travel along the sine wave (always L distance apart), the line connecting the two points at some point has a slope of zero regardless of the value of L. The key is recognizing that at some moment, the left point has a y-value of -1 and at that moment the slope of the line leading to the right has a slope >= 0. And at another moment, the left point has a y-value of +1. At that moment the slope of the line leading to the right point has a slope
and it all works since the slope as a function of x is continuous!
cool motivation... Here is an interesting question... f continuous everywhere implies that f satisfies the intermediate value theorem (for every interval)... does a function that satisfy the intermediate value theorem (for every interval) has to be continuous everywhere?... It is also interesting that mathematicians took the intermediate value theorem for granted (meaning that they did not care about writing a proof for it since it was kind of obvious) until Bolzano in 1817
Haha cue Conway; I also love how derivatives satisfy the IVP, regardless of whether they are continuous or not
Imagine a function that jumps horizontally but not vertically. This would satisfy the intermediate value theorem but not be continuous
@@wabc2336 🤔 do you mean something like the constant function 1, but only defined on, let's say the intervals [0,1] and [2,3]? so it jumps from 1 to 2 horizontally?
I should probably have been a bit more specific and say that a function defined over an interval, and such that it satisfies the intermediate value property in every subinterval of that interval...
In any case, thanks for adding to the conversation!
"[D]oes a function that satisfy the intermediate value theorem (for every interval) has to be continuous everywhere?"
No. See Conway's base-13 function, which sends every real interval to the entire real line through clever manipulation of the "decimal" representation of base 13 numbers. It is discontinuous everywhere.
indeed... that was the @kindiakmath clue in his answer (; It is just interesting to think about it because IVT and continuity seem equivalent
0:25 It does have a formula in terms of Bring radicals, a cool topic people should know more about!
Yep when I was editing I realised there was, though it is still a little complicated... :P
I have no idea what Bring radicals are. However, it is well known that roots of quintics can be found with elliptic functions.
Of course, here the question is, what do you call a "formula"?
HAHAHA a formula is a set :))))
@@rainerausdemspring3584 Probably a function with a closed form expression.
@@soyoltoi This is nonsense. In mathematics everything has a "closed form expression".
“IT” eration
potato vs potato
4:21 It's *Kakutani's
whoops
off the top there's.... this is IVT?
also
MVT
EVT
yeah then there's FTOC but there's only 3 value theorems if i remember cuz' im doin ab calc this year
Yep! And there are implications too; am still contemplating on how to make a video on them similar to this one 😅 since there are some interesting ideas and even applications…
I feel like this still relies a bit too much on visual intuition. Lub proof seems like a better way to do it.
Was trying to highlight the applications of IVT in this video; for the lub / supremum approach it’s in the document attached~
what was used to animate?? Great job! whered you learn to do such animations
ManimCE! It’s the same essential program that 3Blue1Brown used; I learned the basics of Python at university year 1, then experimented with Manim throughout 2023
I thought this was a vid about self improvement from the curve in thr thumbnail 😂
I think you're missing monotonicity as a condition as you don't want repeat intermediate values
Eh I used monotonicity for animations to be simplified, of course in general there could be more than one input for some intermediate values.
Kindiak....Krakowiak
Haha there is a backstory to “Kindiak”
Isn't that trivial?
Haha surprisingly not; the linked document elaborates on the details
BuT iT sAiD i WoUlDn'T BElieVe hahaha.
HAHAHA yes my fellow ProZD viewer