You forgot to age restrict the video

Professor: "This subject is called real analysis" Students: "Finally we meet numbers" Professor: "But we will rarely see any numbers here"

Blackpenredpen: Real analysis Also Blackpenredpen: one plus one is two

'adult calculus" is so accurate. i'm dying in this class right now and most people are dropping like flies lol

I loved math in high school, so much so, that I decided to enroll in math, although I wanted to be an engineer since I was literally three years old. My high-school teacher gave me private lessons and would teach me all about complex numbers, linear algebra, multivariate calculus and I had unbelievable fun trying to prove all kinds of (I guess very simple things) in geometry etc. Doing math really gave a me "rush". Once I started at uni (world-famous ETH Zurich in Switzerland) real analysis "and friends" started and all my excitement about math got flushed down the toilet. I saw nothing fascinating about epsilon-delta proofs, mountains of inequalities and discussing convergence of series until you end up in the ER. I was out of my depth and bored at the same time. A deadly combination. It simply didn't tickle the part of my brain that math did in high-school. I finished the whole basis year, just to prove to myself that I could do it, but I switched to engineering, as I always wanted and all was well again. The joy in math returned, I literally remember I got shivers down my spine when variational calculus got introduced and I spent all night programming my own little finite element solvers and the "rush" was back, with an extra booster. So it turns out I do like math, kinda, at least applied math. I use elements in differential geometry to describe the deformation of elastic continua and I like to believe that I have a good (enough) understanding of all these fascinating concepts, even while having forgotten 99% of all the inequalities and epsilon-deltas. To each their own, otherwise, things would be boring, I guess :)

wow, im taking a formal mathematical proof writing class, and im surprised im actually able to understand this video entirely.

Additional questions: Give an example of the following Q1. f so that f(x) goes to inf as x goes to inf and f is uniformly continuous Q2. f so that f(x) doesn't go to inf as x goes to inf and f is uniformly continuous Q3. f so that f'(x) doesn't go to inf as x goes to inf and f is uniformly continuous Q4. f so that f'(x) doesn't go to inf as x goes to inf and f is NOT uniformly continuous **answers in the first reply**

I’m a PhD in math but enjoyed the video so much. Very nice way to explain

Rather than guessing and checking, there is a systematic approach that you can take here. What you need to be true is |x - y| < δ and ε < |x^2 - y^2|. |x^2 - y^2| = |x - y|·|x + y|. The latter implies ε/|x + y| < |x - y| if at least one of x, y is nonzero. Thus ε/|x + y| < δ. Now, the idea is very clear: choose a real number ε > 0, and real-valued expressions x, y that depend on δ in some way, so that ε/|x + y| < δ is true for any δ > 0 AND |x - y| < δ is true for any δ > 0. The choice ε = 1, x = 1/δ + δ/2, y = 1/δ works, because then ε/|x + y| = 1/|(1/δ + δ/2) + 1/δ| = 1/(2/δ + δ/2) = δ/(2 + δ^2/2), and it is indeed true for any δ > 0 that δ/(2 + δ^2/2) < δ, because 0 < 1/(2 + δ^2/2) < 1 is equivalent to 1 < 2 + δ^2/2. However, you can also realize that |x + y| =< |x| + |y|, so 1/(|x| + |y|) =< 1/|x + y|, hence ε/(|x| + |y|) =< ε/|x + y| < δ, while |x| - |y| =< |x - y| < δ. Thus ε/(|x| + |y|) < δ and |x| - |y| < δ. The first inequality implies ε/δ < |x| + |y|, and this is where, rather than just guessing arbitrarily, you may come across the idea that x and y look something like A/δ + Stuff, inspired by the fact that, on the left hand side, you have ε/δ, while A/δ + Stuff - B/δ - Other stuff can be made to always be less than δ if A, B, Stuff and Other Stuff are chosen wisely, which is actually easier than choosing x and y with very little information. You may even choose x and y so that they depend on ε as well, and it will make the satisfaction of the inequality that much more obvious. For example, you can, without loss of generality, imagine that |x| + |y| = A·ε/δ + B, while |x| - |y| = C·δ, as long as C is restricted to be in the interval (0, 1). This is just a system of equations to solve, and by adding the first to the second, you get 2·|x| = A·ε/δ + B + C·δ, and by subtract the second from the first, you get 2·|y| = A·ε/δ + B - C·δ, so ε/δ < |x| + |y| = A·ε/δ + B, and |x| - |y| = C·δ > δ, which guarantees satisfaction of the inequalities if A = 1 and B > 0. The video makes the implicit choice C = 1/4, B = δ/4 > 0, and the explicit choice ε = 1, but it can have been any other choice while still satisfying the inequalities. The point of this demonstration is that you can make appropriate choices for ε, x, y in full, powerful generality, without having to do any guessing and checking, by simply using the inequalities already given, the properties of the absolute value, and some basic algebra.

"This guy is delta's square over four, I don't care what delta is anymore " As simple as that. Btw thanks for calling us tough🙃

Me: *Sees adult calculus* Me: Well maybe I might be able to und- Bprp: "There are no numbers" Me: Nevermind

"I thought real analysis meant real numbers but there's no numbers" I died.

Thank you for this. I've always wanted to do a degree in mathematics (I got a degree in accounting because I was too afraid to do mathematics), but this has shown me that the maths done in a degree is scary, but not insurmountable. Always love your videos.

I thought people were joking about the fact that numbers disappear in more advanced math.

I'm really liking the higher level math videos lol it's like your content has evolved as I have grown

The funny thing is that while I understand the definition perfectly, I still have no idea what uniform continuity is. 😂😂😂

Is it just me who doesn't understand anything but still watched it because it looks like he's writing newly found powerful spells to enhance his pokemon in order to dominate the Earth.

alright, another video that will go in my "watch later" playlist that i don't know when i'll watch.

Teacher: "Is x^2 continuous, and prove your answer." Me: "Of course its continuous, how did you get to being a maths teacher and not know that?"

I wish you released this video a year ago, it would've helped me realise I'm really not into doing Maths at university/college

You look like a Chinese sage, the ones with the long opium pipes and a beard that could make Santa Clause jealous

Bprp: There's absolutely no number! Zero: Am I a joke to you?!

Uniform Continuity is studied slightly later on in real analysis. My real analysis course started me with point-set topology - Compactness, Completeness, Cardinality, Continuum Hypothesis, Boundedness, Limits and Sequences, and then functional analysis (which includes continuity). Remember guys, a function may not be uniformly continuous; however, a compact set within the domain of a function will most likely be uniformly continuous.

No numbers? I mean, there is "zero".

Doing maths in university in the UK and I just wanted to say I love watching your videos, they never fail to get my noggin joggin!

One formal way to make more obvious the distinction between continuity on a set and uniform continuity on a set is to change the δ quantifier into a quantifier over a function. For example, the definitions in the video are equivalent to the following: f is continuous on I iff there exists some g : R+*I -> R+ such that for every ε > 0, and for every x in I and y in I, 0 < |x - y| < g(ε, y) implies |f(x) - f(y)| < ε. f is uniformly continuous on I iff there exists some h : R+ -> R+ such that for every ε > 0, and for every x in I and y in I, 0 < |x - y| < h(ε) implies |f(x) - f(y)| < ε. The disadvantage of this alternate yet equivalent formulation of continuity and uniform continuity is that proving the negation is not quite as simple or intuitive, but the advantage is that this make obvious what the distinction is between the two properties and why the latter is stronger: in the former, δ, imagined as the output of a function of ε, cannot also be a function of y, the points in the domain of f. This is a further restriction that makes this property a stronger property, a stronger condition, and hence why it is different from simply being continuous.

We're adults now. Me as a teen: *sniffs*

when I was told in my school , that a continuous function is ,when we draw a function's graph without lifting up pen, and now by the theorem u told that if distance between 2 points is less than epsilon then it is continuous , I guess now I can make sense. Btw thank u so much for ur work .

Bprp: this is adult math Adult math: 13:18

Those 6 dislikes are from engineers watching the wrong video, most probably.

My happiness is uniformly continuous now!!!

7:35 Bprp: "there's no number" Zero: "Am I nothing to you?"

I was waiting for a video like that 😌😌

4:18 you could have been working with a random given closed interval [a, b] if you let delta being epsilon over 2|b-a|

I hope I can understand this once I take this course. Just started cal 2 so, still got long way to go.

Great video, but I would have liked you to illustrate this point by picking an epsilon and plugging them into the function to show that you can say "ok, I want my function outputs be be at least 1 apart regardless of a choice in delta. If you give me a large delta, I can make x and y far apart so the output f(x) and f(y) are at least one apart, but if you give me a small delta then x and y are large values but also nearby one another, which means their outputs get stretched along the parabola. I like an illustration of what we mean by uniformly continuous is helpful at the end and how the finite interval prevents you from making extreme choices to break the definition, but an infinite playground for choosing x, y and epsilon allows you to break the definition.

ok, hear me out.... you should do a real analysis marathon where you do several proofs. preferably before the fall semester (when i officially take the course) 😅

That moment if the only numbers on your exam are the date and page number.

This was a great video! I've never really tried looking into analysis, but I can tell that I'd still roughly understand this if I hadn't taken a course on formal logic. (And I definitely could've done that proof if I tried hard enough.) The topology way of thinking of the uniform continuity described in a comment somewhere also makes me understand the concept a lot more than a formal definition does, though you definitely still need that definition.

That incredible feeling of writing the box after a hard work. Can you please do a Taste of Complex Analysis?

*hell yeah! real-analysis let's goooooo*

MAN YOU CRACK ME UP HAHA, THIS IS WAY ABOVE MY LEVEL BUT WATCHING U GET WAY TOO HAPPY IN A LONELY ROOM STARING AT THE CAMERA SMILING IS JUST.. ART IDK, KEEP DOING WHAT UR DOING MAN, LOVE IT

Took two years of honors math long ago as part of a computer science degree, and I joke that the only numbers we saw were e, i, pi, and the page number. (Not that much of a joke actually.)

X=r+delta, y=r Calculate (r+delta)^2-r^2 For large values of r we can understand what will happen to the difference. This is a difficult barrier for many students. If a ten year old kid can understand this he can become a great mathematician.

My approach to finding a delta is that I choose delta as 1/max{f'(x) | x ∈ I} (if it exists). It's a bit like cheating but if it works out...

BPRP: "Wow, absolute no numbers" Zero: "Am I a joke to you?"

"Can you please make up your mind please? please?"

These videos are great. It was like watching a courtroom drama where the prosecution attorney suddenly pulls a damning piece of evidence out of his/her briefcase. Case closed!

You are literally changing my future ❤

So much fun watching the Content and you enjoying what you're doing.

- As you can see, there's no numbers. - (Number zero): AM I A JOKE TO YOU?

I would be so happy with more rpbp analysis videos!

If x ≈ y (in other words, x − y is infinitesimal), then x² − y² = (x−y)(x+y) is also infinitesimal, as long as x + y is finite. So we have continuity at every finite point, and uniform continuity on any finite (meaning bounded) interval, but not necessarily at infinity or on an unbounded internal. And in fact, if x is infinite and y is x + 1/x, then y − x = 1/x is infinitesimal, while y² − x² = 2 + 1/x is not.

Clear explanation… thank you so much…sir

7:32 when someone realizes math is not about numbers

Sir before your video I never thought that mathematics could be so interesting 😀😀😀

Please more proof based videos, they' re great

Please do some videos about Real Analysis. Your style are quite good. You are a natural Teacher.

Another way would be to find two sequences x_n, y_n s.t |x_n-y_n| -> 0 but |f(x_n)-f(y_n)| -\-> 0 here x_n = n and y_n = n+1/n would prove that x^2 isn't uniformly continuous in (-inf, inf)

I'm five years old . My friends are learning basic arithmetic but I can't wait to learn calculus so I'm here .

Real analysis? Adult calculus? Uniformly continuity? Now we're talking!

This brings back memories from the advanced calculus (basically real analysis) that I took years ago!

7:44 blackpenredpen: There is no numbers at all... 0: Am I a joke to you?

Blackpenredpen: there is no number! Zero: Am I a joke to you?

The attempted captions on this video are hilarious.

Idc how hard this is im finishing/passing this no matter what it takes.

Those kinds of proofs we are expected to do in germany in the first semestar. (Cs, math, physics as far as i know...)

3:37 . It is because it's a conditional statement and not Biconditional

Now, this is the mathematics I'm waiting for, thank you sir

It would have been interesting if you had reflected on why the proof for not u.c. in R would not have worked on an interval [a,b]

"We're not doing math for fun today" "BUT BUT: "

Great video🔥🔥it reminds me of my first year in college

I'm an engineering student and it's really frustrating dealing with real analysis.

Me in Calculus II rn: 😳 Adults in Real Analysis: your time has come 😈

I am now afraid of my next semester, I get this class, abstract algebra, number theory, and pdes

Why easy when you can make it complicated! But you rocked it!!! Nice in Corona times,will show it my niece!looking forward to the expression in her face...😘

Why is this man holding a Pokemon ball ?

I wish I had you for Real Analysis some 45-50 years ago instead of Old Fishface, who taught by the stream of conciliation method. Fishface Shen Ji Beng. Ta yeh shi shao ren.

During the class: Test In bed - My Brain: You got log(x)sin(x) for #5, right? In bed - Me: Yes In bed - My Brain: YOU FORGOT +C

I remember taking this when I was in uni. I was clueless for the whole sem lol

As an Electrical Engineering student from israel, i'm quite shocked so many people are so frightened... my calc 1, 2, 3, complex analysis and even fourier analysis during my first and second year were exactly like that. Are you all saying you didn't learn calc like that? Did ur courses contain only calculating excercises and zero proofs?? O.o

Where does the proof for not uniformly continuous use the fact that the interval is R, why wouldn't this proof work for an arbitrary interval [a,b]

Take me back to math methods D; I wanna go back

This was surprisingly easy to follow

Awesome! Excited to see some analysis material!

I do really enjoy all your videos, I've learned too much...keep going 😁. Greetings from México

Please make a "world record video" of solving multiple "JEE Advanced/GaoKao Maths Papers" on streak just like those long calculus videos !!!

That really did feel good to go through and understand the soln.

Just beautiful ... Analysis is more interesting than others

It would be clearer if you either used a for both or y for both.

shouldnt it be the case that: f'(x) is bounded f(x) is uniformly continous ? bc that is way easier to show than constructing an epsylon proof atleast in most cases

This isn’t related to the video but I suggest if you ever want to talk about electronics, you should call the channel “blackwireredwire”

Omg you blew up! Upward bound

You are about as clear as mud!!!

5:34 you forgot to make it non-zero. It should be 0

My basic real analysis prof assumes we know everything already and is teaching it like his 5000 level real analysis class, I have NO idea what’s happening and I’m not passing this class

I love the shirt: QED which originally mean Quod Erat Demonstrandum. (From Latin, something like: The problem is solved) In Israel (Hebrew) we write מ.ש.ל (מה שהיה להוכיח) With the same meaning.

Since x^2 is continuous it would not occur to me that it is not uniformly continuous on R, so how do you know to start the proof that it is not? That is, could you show how attempting to prove that it is U.C. on R fails?

Thanks, dude, I wanted this Love your videos

Title: *(adult calculus)* Video: _why is he holding a pokeball?-_