Day 127 – Practicing Math Live – Ch. 1 Limits

[00:00:00]Hey, [clears throat] hey. What's going on, YouTube? How's it going, Twitch? We should be live now. What's up? Today is uh I think the last day that we're going to be doing calculus for a while. So, farewell calculus. Um we are focusing for the high school math. We're focusing on this other really wonderful yellow book, uh basic mathematics by Serge Lang. So, I'm replacing calculus in the middle school, high school. Well, really, calculus is high school. Uh, we're replacing that with this middle school, high school book, this pre-calculus book instead. Um, and so this week I've been treating college, I work one week at a time. Uh, one week of high school math, which we did basic math in, one week of college math, and then one week of graduate level math.

[00:00:54]So, beginning tomorrow, uh, how was your lunch? Lunch was good, man. Uh, my wife gave me her leftovers from yesterday from this Korean place, and it was so good. Um, hey, Divium, good to see you again. Good to see you. Uh, welcome back. Um, okay. So, farewell calculus.

[00:01:13]Uh, good riddance. Good riddance. Truly, not really. We want you to study Olympiad style maths. Um, not not anytime soon. I just want to be clear. I I want to make sure that we're working systematically, predictably through a few books so that we we know, you know, uh that we're making proper progress.

[00:01:38]Someday I would like to do Olympian math, but uh not soon. Good to see you, though. Good to see you, dude. How was the rest of your uh your lecture? Did you already finish that video, Leo?

[00:01:50]Shout outs to Leo in the chat, by the way. Um, yeah. Hope that video was uh pretty good. I Oh, now verified stream.lealkc.org shows everyone listed is live when live.

[00:02:04]Dude, that's awesome. I want that so much. I would pay you for that if you charged me. I would like, dude, like, let's just build all the missing live stream features that YouTube is missing.

[00:02:19]Let's just build those. Let's just vibe code those and make a bunch of money so that we can just study math on the internet, dude. Like seriously, there's just so much that's missing. And uh yeah, like just a decent like now live list that doesn't require you to like look at your home feed all the time.

[00:02:42]Like I only want to see and check if any anybody's live streaming, right?

[00:02:50]Like I do not want to get sucked into the algorithmic black hole of video recommendations, you know? We'll put up a Discord bot for it soon. Actually, that sounds really dope, dude. Seriously, this is one of these things that's been in the back of my mind. Like, if I ever get around to it and feel like vibe coding, that sounds like a weekend project or a dayong project. Like I bet Claude could like oneshot that app and it would be something that I would be willing to pay for. I think a lot of people would be willing to pay for that.

[00:03:26]Good for you, man, for actually doing it. Okay, so um we finally hit one of all the different types of problems from um section 1.3.

[00:03:37]Uh so you all know there are a few different categories of problems in Tolman and cone which I love. There's the thinking back problems, the concepts problems, the skills problems, applications, proofs and then thinking forward. So yesterday now we have one of every type of problem done from sections one, two and three. now.

[00:04:06]Yeah. Uh, Beetle, your name is Beetle. Leo was just reacting. Leo was just watching that uh that video on his stream. By the way, shout outs to Leo uh Leo streams if anybody doesn't already follow.

[00:04:24]You should follow Leo for real.

[00:04:28]Um I I look forward to to watching that video someday. I don't know when but it's a dude I don't know Leo like this is a question for you man like foras theorem is one of these monumental beautiful results that has fascinated me and I want to learn and I think I could learn it properly sometime someday right doesn't it feel like it's like kind of robbing that result from me to not learn it from a textbook like I kind of I kind of want to take it slow and work the exercises and get the history and work through books to to you know climb that Mount Everest and learn it the slow way not the like here's the you know snappy 5 minute highle summary like yeah I don't know man I don't know I don't think the video's explanations are that good see like here's the thing I don't trust this person when I saw that video I I thought it was AI at first Like at this point you you don't really know like it could be a deep faked AI video and I don't really want to learn from Monserum from from a deep fake AI math YouTuber.

[00:05:46]Like dude, come on. I mean I he probably isn't, but soon my default reaction is going to be YouTuber is probably AI unless I've met them on Zoom, you know.

[00:06:00]Um, again there's a question of to what degree do you want to know it? Yes. Uh, not aware of a textbook account of FLT.

[00:06:08]Actually, well, actually, my friend Arod, I'll show you.

[00:06:20]I really want to work through both of these books. As I understand it, these Edwards books are very good at giving the history like the historical context of um Gawa theory, how it was invented and then oops like introduction to FLT.

[00:06:42]This is really introduction to algebraic number theory. Um I mean this was published I believe in the 80s or 70s or something. So obviously this is this does not go through um you know Andrew Wilds's proof but it gives the historical context that gets you thinking about the problem I suppose like this one looks really cool. I I've been wanting to work this one for a while. This draw guy is never touching this book. Which one? This.

[00:07:15]Yeah, it would I don't know. We'll we'll get there someday.

[00:07:19]Um, you know, mate, but you're stuck on the Mount Everest of doing the same linear algebra exercises 10 times a day. Funny enough, we've moved on from that. We're not going to be working that linear algebra book anymore. I am actually replacing uh linear algebra in my college series. I'm replacing it with abstract algebra next. So, I'm excited, man. I hope you all are, too. Yeah, this book looks really good. And then there's a lot of newer books. There's there's a bunch of newer books as well. But I want to I want to understand algebraic number theory, you know, in the context of for theorem and like why we understood it to be a very hard problem and you know some partial results and things like that before I just jump right to the to the conclusion or something.

[00:08:20]Slow and steady, dude. Slow and steady.

[00:08:23]That's the That's the key, I think.

[00:08:28]Yeah. I mean, I think it looks good. I' I've read very good things about it. Um I've read very good things.

[00:08:37]Uh now we're doing high school calculus thrice a day. Today is like the last time. And did calculus is is college.

[00:08:45]It's like a good, you know, last class of high school or first class of college. This one counts as college math this week and we're done with it. Um, continuity. We're finally moving on to section four for the first time. Uh, let me jump all the way down. So, now let's get started. Section four on continuity. All right. So pro I always go straight to the problems.

[00:09:15]Okay. Exercise.

[00:09:20]I want to do first a thinking back problem first. Thinking back.

[00:09:28]Uh number one thinking back finding roots of peacewise defined functions for each function f that follows. Find all values f of c for which f of c equals z. Check your answers by sketching a graph. Okay. So our function is uh peacewise defined as 4 - x^2 if x is less than zero and it is defined as x + 1 if x is greater than or equal to zero. And it asks find all C such that find all roots of this function such that f of c is zero and then sketch the graph.

[00:10:36]Okay. So solution to this uh if f of c equals 0 then uh c is less than zero and 4 - x^2 = 0 or c is greater than or equal to zero and x + 1 = 0.

[00:11:14]In the first case, um c is less than zero and x2 = 4 has one solution.

[00:11:38]uh which is x = to -2.

[00:11:47]In the second case, x + 1 equ= 0 and x is greater than or equal to zero has no solution.

[00:12:10]as x + 1 = 0 implies x is -1 which is less than zero. So we need both the equation to be true and the condition to be true. So x in this case is not greater than or equal to zero. Um, thus exactly one.

[00:12:43]Do we call do we call zero points roots even for like peace-wise defined functions or is like the word root somehow limited to polomials or something like I think that we we still call them there's zeros or roots. There's exactly one zero of f ofx which is f of uh -2.

[00:13:23]This is the only zero of the function.

[00:13:26]And the graph reminding ourselves that our function is defined peacewise as I believe 4 - x^2 when x is less than or equal to 0 and x + 1 when x is greater than 0. Was that correct? 4 - x^2 x + 1. Yep. Um, so when x is greater than zero, I'll do the second piece first. It's a little easier to see, I think. Um, the y intercept, uh, like it it has a limit point of y = 1.

[00:14:08]Um, and we'd have this 45° angle continues. Um this is x + one plotted on the right half of the plane.

[00:14:23]Um and then on the left half uh 4 - x^2 1 2 3 4.

[00:14:34]So when x is -2 um y is zero the function would look something like this.

[00:14:59]This point is solid and then this point is an open circle.

[00:15:07]It's a little hard to see, but here's approximately our peace-wise function.

[00:15:12]Everything working? 01 should be open.

[00:15:16]What's going on?

[00:15:22]Hey, Baldwin the anchorite. I'm happy to hear that you're enjoying the the videos and the problems so far. I think that like good for you, dude, because the real learning is going to happen with the exercises. It's always going to happen with the exercises and uh I think that they're fun. I think Lang's book is really well written and accessible. I don't need to like walk through every single exercise and and step through like everything. Not yet. like maybe when things get a little harder, but I'm trying to encourage those who are who are following that series to actually read the section because that's when that's how I'm going to help people develop that important thing called mathematical maturity. Like the ability to read a book on your own. Um, okay. Let me catch up in chat. So, we just finished this one. Heck yeah. I'd call them zeros, but if you said roots, I'd understand what you mean. Yeah, after I said roots, I was like, I think zeros is just a little more accurate somehow.

[00:16:31]Yeah. And roots are for polomials. I'm glad I was thinking that. I I I'm glad I was thinking that because I was like I think you you reserve roots for polomials, but then you speak of zeros for other for other functions. Thanks for confirming y'all. I just want to get this open circle a little more visible here.

[00:16:55]Okay, so here we go. This function has exactly one zero at um -2.

[00:17:10]Here's the only zero of that function.

[00:17:15]Um, yeah, we got a bunch of self-studyers in this chat, in this community. That's freaking awesome.

[00:17:23]Um, that's the goal here, dude. I think so many people are studying math for fun and there it's just under represented.

[00:17:32]So under represented.

[00:17:35]Um, who are you all talking about? He brings up GCD. Who is he? the person in that video, the FLT video.

[00:17:48]Yeah. Like if you're going if that FLT video introduces GCD and begins with like GCD and gets all the way to FLC, this sounds like an amazing video.

[00:18:00][laughter] That sounds awesome.

[00:18:01]Actually, um yeah, TBH, the context of Formos theorem is nearly all of number theory, right? Right. Uh yeah. Yeah.

[00:18:11]Yeah.

[00:18:13]FLT motivates so much of number theory, right? That's why I want to learn from a book, you know, that like guides me through all the other stuff, not just FLT and not handwaving. I want to I want to get the whole path there.

[00:18:34]Um, yeah, Beetle, like that video is going to move fast. I think like if you're not if you don't have any number theory background, um I just I would I would wait like you're not going to you're not going to be able to follow somebody trying to prove FLT like motiv like watch it enough to motivate you to crack open a number theory book because you got a lot to learn first.

[00:19:08]Um, Baldwin, super happy to hear that you're enjoying it. 01 should be open.

[00:19:17]Um, yeah. I mean, this this function is defined on the real numbers. So, I think when you said open, you mean like to use a like an open circle that it's not included that point. Um, and that would be true. Oh, yeah. Yeah. 01. Yes. Yeah.

[00:19:36]Yeah. like the coordinate x= 0, y = 1 is open. Yes, I I clarified that as we went. Yeah, good catch. You were right. You were right.

[00:19:52]Congrats, Apex Goblin, for passing the for dominating the J. That's awesome, dude.

[00:20:10]Yeah, I'm super curious what it's like for you trying to watch that video, Beetle. And Beetle, if you don't know, Leo just did a video talking about that video that you're watching. So, you should definitely watch his his stream, his video from earlier today, I think.

[00:20:29]Um, Leah lectures is definitely making videos on number theory, I think. No problem learning from this video. Okay.

[00:20:37]Okay.

[00:20:39]Uh, Erdos unit conjecture recently.

[00:20:49]Yeah. Complete playlist with proper Olympiad style problems.

[00:20:54]Algebraic topology. I want to learn algebraic topology. I never have. Okay, cool. Well, we did the first one. We did a thinking back problem and this is now setting the stage for something called continuity. Um, and sneak peek for those who don't know, this function is discontinuous because the limit at zero from the left does not equal the limit from the right.

[00:21:33]So at that point you can find uh an epsilon delta neighborhood you know where you can always um I don't know find y values that are outside of the neighborhood like yeah the limit is not defined at every point I that for a function to be continuous its limit needs to be defined at every point.

[00:22:04]Um, so let's start reading. Let's start rereading and learning the section problem zero under concepts in this book. There's a nice pattern.

[00:22:15]They they spend so much ink writing this in every single exercise set. Concepts.

[00:22:22]Uh, problem zero.

[00:22:25]Read the section and take notes.

[00:22:31]Every single exercise set reminds you don't forget to read the section and um read the section and make your own summary of the material. Problem zero.

[00:22:41]All right, let's do it. Um I've learned this before, so I'm going to try to take a highle approach through this.

[00:22:52]Just going to be reading um section 1.4. for continuity and consequences has three main topics.

[00:23:17]We've got uh continuity of functions at points and on intervals and basic types of discontinuities.

[00:23:30]Continuity at points and intervals um and simple discontinuities.

[00:23:43]It says basic types of discontinuities.

[00:23:46]I'll say simple discontinuities.

[00:23:54]Um then secondly we have oh simple functions that are continuous on their domains.

[00:24:05]Simple functions continuous on their domains.

[00:24:16]Uh and then thirdly the extreme value theorem and the intermediate value theorem. So this section we get to extreme value theorem and intermediate value theorem.

[00:24:29]And I know I know what most of these words mean equals the limit from the wrong. Yeah.

[00:24:38]Uh, can I Well, the math actually the iPad the right hand side of the iPad is um is showing like the the way that the iPad shows up. It wouldn't really help that much more actually. Like it wouldn't buy very much more screen real estate. Um, this is already like as wide as it as it gets.

[00:25:08]Otherwise, I would be chopping I'm already chopping off some stuff at the bottom of the screen.

[00:25:15]Oh, well actually not as much as I thought, but yeah, the we're more bound by um the height of the iPad than the We're more bound by the height of the iPad than the width.

[00:25:37]I suppose maybe I could take back some more from the top, but Anyways, I played around with it a lot and tried to I found that this is a a decently good aspect ratio. Um, why did I put my streams on members only? It's because people keep watching the stream and then unsubscribing.

[00:26:01]Uh, people don't realize that the streams are not meant to be rewatched.

[00:26:06]Like the streams are actually literally meant for the people who are there live.

[00:26:13]Like if you want to watch the streams afterward, that's awesome. But uh I kept getting comments from people that say, "Dude, you're spamming us." Like, "Dude, like chill out. Why are you doing so many different subjects?" It's like, "No, actually like uh this is this is my own study sessions over like a year.

[00:26:34][laughter] The like I didn't just film these and I didn't just post these." Um, yeah.

[00:26:43]So, um, I mean, there's that and then it's it's a way that, um, y'all can help support what I'm doing. Obviously, what I've landed on is I'm thinking that the most recent week of video of live streams for any subject is free. And so, there will always be fiveish videos for every subject. Um, and then you can check those out. But if you want to go back and watch hundreds of videos, um I I think it's worth the the lowest tier. I mean, the Luna level membership is $1 US per month. It's the lowest price that I can set or 59 um rupees.

[00:27:33]Like it's not very much. I'm sorry. Hey, thank you for somebody for the for the 550 jewels, bro. Thank you, Luke. Thank you, Luke. Uh, so much. Let me put you on the thanks page for that, bro.

[00:27:50]Appreciate you. And I'm going to give you credit in uh in this math um exercise set, bro. I appreciate.

[00:27:59]Should I write the whole name? That's a long name, dude. Uh, Luke, let me know if you want something else, but Luke, I'm just gonna go Luke Downs. It looks like you wrote that a few times. Luke Downs. Luke Downs. 5900.

[00:28:14]If you want, I could add the 5900 if you find that important, bro. Appreciate you.

[00:28:22]No, no need. I have no idea what it is.

[00:28:24]Uh, this is just my thanks page. Thanks to everybody who's a channel member or anybody who's ever tipped anything like, thank you for, you know, your support. I appreciate you all. Um, and and I will also put you right in here.

[00:28:49]Where is it? Exercise set.

[00:28:53]Thank you for supporting calculus.

[00:29:02]Luke DS.

[00:29:08]All right, dude. You're right at the bottom of my textbook on that page. Uh, I'm doing well. What's up, technical sudep? How are you? How are you?

[00:29:18]Appreciate you. Thank you. Um, and I'll I will hydrate. Absolutely.

[00:29:25]Beetle, you've never seen a good math teacher IRL. That sucks. Uh, not going to lie, I saw you connections and assumptions video yesterday. Well, baked AF and it was actually good how you weren't forcing us a single perspective answer, but we looked at math from an overview. Hell yeah, dude. Hell yeah.

[00:29:44]Um, I don't know. Connections and assumptions video. There's there's a lot that that could describe, but I'm happy to hear because like math really is an offshoot of philosophy. Like there's like there is a lot to math. Like it is not just like a black and white. This is the one true math. This is the one true way or something. There are like you need to understand what your assumptions are and agree on the same assumptions and then you can agree on results. But different people describe different, you know, start from different assumptions.

[00:30:27]Do you skim over mathematical exercise on streams? Yep. I stream almost every almost every weekday.

[00:30:35]Um, why didn't you do an MSE or PhD in math and then research? I was going to, but then I I wanted to make money instead. Uh, I wasn't I felt like I wasn't good enough to to become a math researcher. I love it very much, but it just it's a hard path. It's a hard job.

[00:30:53]Um, instead I I did my math bachelor's degree.

[00:31:00]Uh, and then I worked as an iOS software engineer at Apple.

[00:31:07]Um, here I'm going to do it this way.

[00:31:09]Apple, Lyft, and Google.

[00:31:13]as a software engineer.

[00:31:20]Then uh my entire team got hit by layoffs shortly after ChatBT came out. Uh, so ever since I've been an independent entrepreneur, creator person, like a YouTuber, creator.

[00:31:44]Uh, and next I want to become a freaking school teacher. I want to teach the little children.

[00:31:55]That's the next goal.

[00:31:57]So, thank you for all for helping me on this uh on this amazing journey.

[00:32:02]Appreciate you all.

[00:32:05]Yeah. Yeah. Goal is to become a a school teacher. Cal 3 is vector calc. Yeah.

[00:32:11]Hey, lock. Good to see you, dude. All right. So, um but I want to continue also at the same time.

[00:32:24]Dude, you can just study math as a hobby.

[00:32:28]Like, I want to learn PhD level math and maybe even do research someday. Like, as a hobby, you can do that. It's not impossible. It's not impossible.

[00:32:42]Um, okay. So, now I know what the topics are going to be in this section.

[00:32:49]Um, yeah, you'll hate on calc 3 because it's got vectors.

[00:32:55]Lock does not like vectors at all. Does not not a fan. Okay. Intuitively, a function is continuous if it's graph has no breaks, jumps, or holes.

[00:33:07]Okay. Intuitively continuous means no jumps.

[00:33:21]Huh? If its graph has no breaks, jumps or holes.

[00:33:27]Oh, yeah. No jumps, breaks, or holes. Okay. So I misspoke earlier because earlier I said I I misspoke because I said that the limit from the left needs to equal the limit from the right but it also needs to equal the function value at that point. So for example, the graph of 1 /x um has a hole at zero.

[00:34:11]This function is discontinuous because it doesn't have a value at that point.

[00:34:18]So like an intuitive explanation is that a graph is continuous if you can draw it without lifting your pencil at all. Like you can draw the thing and it's not that is not the graph of 1 /x. I was thinking that I was like wait a minute that's not right. I was I was thinking that doesn't sound right. Yeah. Yeah. 1 /x. My bad.

[00:34:42]Uh oops. Lol. No, I meant um x^2 overx.

[00:34:50]Yeah, [laughter] my bad. Thank you, lock. Thank you, lock. I blanked for a second. Um, this thing is similar to y= x, but it's discontinuous at the point x= 0. That's an amazing graph of 1 /x. I know. Yeah, I'm I need some more coffee. He just likes unemployment. I know, right? I'm He likes doing PhD research as a hobby.

[00:35:18]That's the goal. That would be awesome.

[00:35:20]If I could, I would. Um, that sounds freaking fun.

[00:35:29]I mean, I'm I have no progress at all toward making toward contributing research like, but that would be that would be fun in in theory. I I would like to do that. I would like to try.

[00:35:42]Loosely speaking, you can sketch the graph of a continuous function uh without picking up your pencil.

[00:35:55]If you can draw the graph without picking up your pencil, um we can make it more precise by using limits.

[00:36:03]Consider these graphs. While the first has no breaks or holes, the remaining three all have some sort of bad behavior.

[00:36:11]And the the examples that they give are y equals well yeah it doesn't um it doesn't say exactly what the f ofx is but it looks like y equals x. This next graph they draw piewise and it's a graph that looks like this.

[00:36:36]Then the third graph is one that looks like this.

[00:36:45]Uh and then the fourth one is a graph that looks like this.

[00:36:53]So all of these have a bad behavior quote unquote. There's a hole, there's a jump and other things. Um turns out that limits as x tend to one detect exactly this bad behavior. In each case uh the limit as x approaches 1 is not the same as the value at x= 1. Um for example um these functions are f ofx g ofx um h ofx.

[00:37:29]They're labeled these h of x and k of x.

[00:37:34]So g of x uh let's see [snorts] limit as x tends to 1 g of x does not exist.

[00:37:54]Um but g of 1 = 2.

[00:38:02]Uh the limit of h ofx = 1.

[00:38:17]But h of 1 = 2.

[00:38:24]Uh and limit of k of x = 1 but k of 1 does not exist is undefined.

[00:38:49]So whereas for f the limit equals the function value.

[00:39:04]So these motivate this definition definition 1.12.

[00:39:16]Next week we will be back to representation theory for our streams but I will be uploading two videos for the basic math series. And in basic math we will be moving on to section 1.4 on multiplication.

[00:39:32]Uh or wait 1.3 on multiplication I believe.

[00:39:38]Yeah, I do not have children if you're asking me.

[00:39:43]Heyi hey, thank you. Huge huge 0701. I appreciate you very much.

[00:39:52]If you don't mind, I would like to write you in my thanks page, dude. I appreciate you. Um, huge. Oh, wait. No, two U's, not two G's.

[00:40:08]Don't know if they're even in the stream or if they just uh became a member for the videos for the VODs, but thank you, bro.

[00:40:18]Appreciate you very much.

[00:40:22]This is kind of a little speed bump to do this every time, but it's worth it. I think P2 is definitely a cool plane. I don't know what that is.

[00:40:40]Ah, I did it again. It should be two U's, not two G's. Okay, whatever. Two U's, G and one G. Okay, I'm going to scratch out that name. Try it again. E U G 0701.

[00:41:00]Thank you, bro.

[00:41:08]Definition one two continuity of a function at a point says A function f is continuous at x.

[00:42:03]If the limit at x= c at a point c if the limit value at c equals the function value at c.

[00:42:25]So here is our definition of continuity at a point.

[00:42:32]Cool.

[00:42:35]That's what I remember. By considering one-sided limits, we can get a more detailed picture of continuity.

[00:42:41]Um, right continuous at x= 1, but not left continuous.

[00:42:47]So definition 1.13 left and right continuity.

[00:43:07]A function f is left continuous.

[00:43:25]If the limit value from the left equals the function value at that point and is right continuous.

[00:43:48]Similarly, same thing.

[00:43:56]Um, okay.

[00:43:59]Sometimes convenient to talk about continuity of a function on an interval.

[00:44:05]So definition 114 we have uh continuity of a function on an interval.

[00:44:29]Uh a function f is continuous on an interval i.

[00:44:34]Function f is continuous on an interval i.

[00:44:49]If it is continuous at every point in the interval of i, right continuous at any closed left end point and left continuous at any closed right end point. So there's three points um continuous at every point in I in the interior of I.

[00:45:26]It says interior of I.

[00:45:32]Um, write continuous uh at any uh closed left end point of I.

[00:45:55]And similarly left continuous at any closed right end point.

[00:46:10]Okay, let me rechhat you can have a I never understood the point of leftright continuous. thought the whole point of introducing continuity was avoiding jumps. Um, I think here's part of it. I think that we want to you can define [snorts] continuity to like on an interval like you it requires right continuity and left continuity. I suppose that this is where um well h you can have one-sided continuity when the function isn't defined past that value. Also, jump discontinuities are probably the most well- behaved sort of discontinuity. So, it's nice to get a good handle on them. Ah so yeah if you define a jump this like this function is continuous on 0 to one and continuous on one to two or something or one to infinity but it's not continuous on 0 to two, right?

[00:47:53]Like this discontinuity is a well behave like one of the most well- behaved discontinuities I think is what Arod is saying here. Yeah.

[00:48:09][snorts] Um, there's another section that says types of discontinuities.

[00:48:42]There's a removable.

[00:48:47]Yeah. When a function is not continuous at a point x= c, we say that is not that it is discontinuous at x= c.

[00:48:56]So um not continuous at C we say that it is discontinuous that's just a definition discontinuous at C three most basic types are a movable discontinuity.

[00:49:33]Um, and the example that the book gives is this with a hole here and a filledin point here or something.

[00:50:02]A jump.

[00:50:08]A a jump discontin continuity.

[00:50:14]BL a jump discontinuity.

[00:50:24]>> [snorts] >> And then an infinite discontinuity. video.

[00:50:55]Intuitively we say that a discontinuity is removable.

[00:51:12]If we could remove it just by changing one function value.

[00:51:17]If we can remove it by changing just one function value.

[00:51:35]Interesting.

[00:51:37]Um at a jump discontinuity the function jumps from one value to another and at an in infinite discontinuity the function has a vertical asmtote.

[00:51:49]Okay. So as these look uh these types of discontinuities can be described pre precisely in terms of limits as follows. So if you were ever asked to write down a formal definition of those types of discontinuities, this is how you would do it.

[00:52:20]removable jump and infinite discont.

[00:52:33]Um suppose f discontinuous at x= c.

[00:52:48]We say x= c is a option one uh removable discontinuity.

[00:53:12]If the limit exists but is not equal to f of c.

[00:53:38]Secondly, our second option is a jump discontinuity.

[00:53:45]Jump.

[00:53:47]That's a horrible looking jump. My handwriting is getting worse by the day.

[00:53:52]Jump discontinuity.

[00:54:00]Uh if the limit from the left um and the limit from the right exist but are not equal. So here's a point where this answers lock L lock lock's question earlier of one of the reasons you would want to define limit from the left versus right so that we can classify that type of discontinuity as a jump discontinuity.

[00:54:40]Uh and then we have infinite discontinuity.

[00:54:52]Uh if one or both [snorts] um is infinite.

[00:55:29]Now we have our formal definition classifying these simple discontinuities and we can uh write it down very precisely using the language of left and right sided limits.

[00:56:00]>> [snorts] >> one-sided continuity, not one-sided limits. Oh, I see.

[00:56:15]I see.

[00:56:17]Yeah. I don't know. I can't I don't remember. Like, as I'm writing all this stuff down, I'm like, I vaguely remember this, but it's been a long time. I don't remember examples where knowing that it's continuous on either side is useful or not. I don't know. Yeah.

[00:56:40]I don't I don't know.

[00:56:43]I don't remember analysis.

[00:56:46]I think it's useful for trivia mostly.

[00:56:50]Um continuity of very basic functions.

[00:56:57]Uh so this is an entire section. I'm just going to trust the textbook author here. Continuity of very basic functions.

[00:57:18]Um we say that a function is continuous on its domain if it is continuous on every interval on which it is defined.

[00:57:28]Continuous on its domain is defined here.

[00:57:38]Um the following theorem proves that unsurprisingly our simplest examples of functions are continuous on their domains.

[00:57:49]Okay, we have a theorem. Finally, theorem 1.6 says continuity of simple functions continuity of simple functions.

[00:58:13]Constant identity and linear functions are continuous everywhere.

[00:58:21]Constant identity and linear functions are continuous everywhere.

[00:58:44]in terms of limits.

[00:58:46]We write out all of the things Power functions are continuous on their domains.

[00:59:21]Here we go. Now, this is a more powerful statement. Power functions are continuous on their domains.

[00:59:33]In terms of limits, if a is real and k is rational, then for all values x= c at which x to the k is defined, we have dot dot dot power functions in terms of limits.

[00:59:51]If a is real and k is rational, then for all um values x= c at which x to the k is defined.

[01:00:17]We have limit as x tends to c of a x to the k power is a c to the kth power.

[01:00:31]Okay.

[01:00:35][snorts] So part B gets us all the poloms does that also get you I don't think you can write x^2x as a power function. That's not a power function, right? That's a rational function.

[01:01:10]It's a ratio of two power functions, right?

[01:01:16]So, we get things like square root of x and others. In probability theory, cumulative distribution functions are specifically right continuous, and that's important for keeping track of atoms in the probability measure. What?

[01:01:32]That sounds cool. [laughter] Neat. Almost all functions are discontinuous. That's That's some crazy talk.

[01:01:40]That is Yeah, it's amazing that we have these kinds of facts. [snorts] The real numbers are very strange, aren't they? They really are. Um I am in part B of the theorem. The limit may sometimes be only one-sided.

[01:02:06]For example, f ofx=<unk> x is defined at x= 0 and to the right of x=0, but not for x less than zero. Therefore, the corresponding limit statement is one-sided.

[01:02:19]Okay?

[01:02:23][snorts] Although it seems graphically obvious that the simple types of functions described in this theorem are continuous everywhere. they're defined. To actually prove continuity, we need to appeal to the definition of limit. We will prove part A here and discuss the proof of part B. After we learn about limit rules in the next section, you will prove that f ofx= x to the k is continuous on its domain for k= 2, 31, -2, 1/2, and negative - 1/2 in exercises 89 to 92.

[01:02:58]Nice.

[01:03:00]Okay. And then there's a proof. I'm going to skip the proof for today because we are moving on from calculus. Haha.

[01:03:10]Extreme and intermediate values of continuous functions.

[01:03:41]>> [snorts] >> We discuss we examine two important consequences of continuity. A con a continuous function on a closed interval must be bounded and attain its upper and lower bounds.

[01:03:55]Okay. Uh second, if f is continuous between two values, then the corresponding values of x go through every possible intermediate value between the y values f of a and f of b.

[01:04:08]I remember these. I remember these. Both of these are uh results of them having unbroken graphs. Um consider a function d. I think I follow that.

[01:04:29]Yeah. Let's write these theorem 1.17 uh the extreme value theorem.

[01:04:53]Okay.

[01:04:55]If f is [snorts] continuous on a closed interval a to b.

[01:05:13]Then there exist uh values capital m there exist values and lowerase m in the interval such that f of capital m is the maximum value of f ofx on a on the interval and f of lowerase m is the minimum. um value of f ofx on the interval.

[01:06:31]Okay, the extreme value theorem. So on a closed interval, a continuous function attains its minimum and maximum values.

[01:06:43]That makes sense to me because we don't it doesn't shoot off in shoot off to infinity.

[01:06:50]Um and it uh [snorts] because it's on a closed interval.

[01:07:26]It's not like it's infinitely getting closer to some suprema value that it never attains. I think I think this makes sense.

[01:07:46]Yeah, I was thinking I do remember this stuff with uh compactness and metric spaces. Hey, what's up Jack Mac? No, this is the study sessions. So, for a long time, I've been a a daily live streamer or at least uh weekdays mostly studying a variety of different math books. Um, the Serge Lang series is a brand new series where every week I'll be posting two or three videos on on on that Lang textbook. But right now is calculus. Today, actually, you're joining the last day of the farewell calculus tour.

[01:08:26]Fairwell calculus. Good riddance. Um, I like calculus. It's just it's not what I would study for fun. Like I I I love calculus. If I have infinite time, I I would I would keep studying calculus.

[01:08:41]But uh for the sake of getting a job or supporting the basic math series, I think it's going to be better to replace my college level math with either abstract algebra or number theory. So I really want to know what you all think like lock and a rod and like the other mathy people.

[01:09:06]Do you all think I should add in like for myself studying and relearning number theory or abstract algebra? Which one would be better overall? Like both of those sound fun to me, but for the sake of introducing like all these new people who are um who are getting turned on to pure math, I kind of feel like abstract algebra. I want to share abstract algebra with a lot of people and abstract algebra I think connects well with representation theory. Um, no I have not checked out the algebraic derivatives thing. It looked really cool. I skimmed it but I haven't I haven't worked through it at all. Um, but yeah, nice to nice to meet you. I think Jack Mack um Sarah Lang that that book is pre-calculus like we're starting at the very beginning but ooh yeah honestly I don't see enough down to earth but also advanced number theory content on YouTube right like I'm thinking like you would vote number theory yeah like I think like basics like like Oilers's toant function and stuff like that which counts the number of relatively prime uh what is it? FN counts the number of wait yeah counts the number of relatively prime integers less than or equal to n. Right? There's some really cool number theory around this. Like you can prove a lot of really cool stuff here. I think that it's multiplicative if I remember correctly.

[01:11:08]For example, um you can get for little theorem you can prove with and I think it relates to this to function for little theorem.

[01:11:25]Yeah, I mean it's it's tempting. I might I might do number theory. I have a first test on Monday and you're in calc one in the fall. possibilities to touch some algebraic number theory. Uh, it's gonna be a while. Yeah, eventually abstract algebra and number theory. I just I don't know, man. I I feel like number theory is just it's a it's kind of familiar and I like the idea of introducing people to a new subject. Although number theory, most people haven't really ever studied number and like number theory is not even required for a math degree. Like at least at my university and I think everywhere, number theory is still an elective that you can take or ignore and I loved it. I took it and I loved it. But um algebra is required. Like algebra you cannot you cannot skip algebra and like the most important number theory you can learn in algebra.

[01:12:33]Are you talking about Gaussian integers?

[01:12:35]Yeah. I mean that would come up if we if we did a number theory stream. We would primes are nice. Yeah. They're they're not ugly at all. Yeah. Wow.

[01:12:47]So this is uh Oiler's toient function, but either number theory or abstract algebra could be a good stream topic to uh to show more proofs, like more intro to proofs. I think I want to do an intro to proofs series for my um college level uh streams.

[01:13:26]Torch. Well, that's what we're here for.

[01:13:28]Maybe we could help you with that.

[01:13:30]They're fun, dude. They're fun. Maybe you just never learned it the right way.

[01:13:36]I don't know. You also fear number theory in a good way. Like you want to learn it with us. like we could help you, you know, we we could help you, y'all.

[01:13:46]Like it's it's nice stuff. It's really pretty. Um, okay. So, now theorem, the intermediate value theorem.

[01:13:57]I didn't write down the number, but who cares? If f is continuous on a closed interval AB, then for any for any k strictly between f of a and f of b.

[01:14:34]There exists at least one I'm going to put it in parenthesis at least one C in the interior such that F of C equals K. Intermediate value theorem.

[01:15:13][snorts] Yeah.

[01:15:16]These two important consequences of continuity may seem obvious, but in fact they rely on a subtle mathematical property of the real numbers called the least upperbound axiom.

[01:15:29]These these rely on an axiom.

[01:15:42]the least upper bound.

[01:16:01]These rely on a property of the real numbers.

[01:16:11]Properly explaining the proofs of these theorems is outside of the scope of this book.

[01:16:18]So this book does not explain the proofs of these but it states them.

[01:16:27]Um, great. This book is not a real analysis book outside the scope of this book.

[01:16:48]In the extreme and intermediate value theorems, the hypothesis that f be continuous on a closed interval is essential.

[01:16:57]Um, closed interval is essential and it gives some examples of functions where this has no minimum value.

[01:17:23]Uh this function has um there is no x value that takes takes the y value k like the function never takes the value k in this one.

[01:18:02]uh the fact that it's continuous on this interval continuity on the closed interval is essential. This second function is not continuous on that interval. And then this last Oh wait, no no C with F of C equal K.

[01:18:42]So this one, the second fails the uh intermediate value theorem. This first one fails the extreme value theorem. And then the last function that they draw, I don't need to draw the axes.

[01:18:58]Just the basic shape and idea of these functions is enough.

[01:19:04]This function is not continuous um at that point C.

[01:19:12]It has no maximum value on this closed interval.

[01:19:23]It's not even defined at x equals c. Dang, you famous. I know, dude. It's crazy.

[01:19:32]Hey, Siraj. Yeah, bro. Bro, how's it going? You don't have to ask a thousand times. It's cuz what I'm doing is valuable, dude. It's worth a dollar.

[01:19:43][laughter] I think it's I think it's worth a dollar. I am um I am leaving up the last five or so of every series. So, you can check out those and if you want to go back and watch hundreds of previous live streams, um, it's just it's set to the lowest value that YouTube lets me put it at.

[01:20:12]So, you know, um, thanks Leroy. Appreciate it, man. Am I a professor? No, I am a software engineer. I'm a math major turned software engineer returning back to math and I'm becoming a a high school math teacher. Yeah. So, I mean I like the live streams are free. The live streams are really meant for those who are who are here [snorts] live during the session. Um but I keep seeing in my analytics that people watch the streams like the VODs and then they unsubscribe.

[01:20:52]I'm getting a bunch of unsubscribes from people who watch the the live streams because this is totally unstructured.

[01:20:59]It's not, you know, like this is not organized. It's not a course. Like a lot of people keep thinking that the streams are the course. And I want to make that clear to people somehow. So, I feel that um putting them out there as you know a soft payw wall is is a good way for me to say like this is not the course. The course is the the basic math videos. Hey, thanks Muhammad. I appreciate the the hundred. Appreciate it, dude. Math wiz, retired software engineer, guitarist extraordinaire.

[01:21:38]I wish I were retired, man. I wish if I could do this for a living, I would do this, man. Like, I I love math. I love classical guitar. I love streaming. All this is fun. Like, if I could, that would be fantastic. But I also just want to teach kids. Like, even if I weren't getting paid, like I feel if I had the chance, I want to just teach kids, dude.

[01:22:04]Is it not?

[01:22:06]I don't know. Um, yeah, somebody correct me. I thought it's Muhammad.

[01:22:10]Is that right?

[01:22:12]All right. Well, anyways, let's keep going. There's the We just wrote down the intermediative value theorem and the extreme value theorem. Um, and extreme value theorem, if this is in Stuart, like I bet it is in Stuart, but it's not tested on the on the AP exam.

[01:22:34]At least back in my day, they they didn't test for this. I think I got two pennies.

[01:22:42]I think it was two jewels. I think you just gave me two pennies. I appreciate I appreciate it very much. Um, reinvent math. You won't. No, I'm not going to reinvent anything. I can reinvent streaming. There's too many video game streamers. We need more math streamers, dude. I I don't think I'm particularly special here. I'm not the best person at all at math. Like, I'm not even close to that. Um, but yeah, we can I think we can get some people to stop playing video games online and start studying math.

[01:23:22]We can we can get people to to switch from Tik Tok and video games to math.

[01:23:31]I really think so. Okay. Theorem 119.

[01:23:38]These beautiful tasty theorems.

[01:23:52]Yeah. A function can change sign only at roots and discontinuities.

[01:24:18]Cool.

[01:24:28]Um precisely it says a function f can change sign from positive to negative or vice versa.

[01:24:53]Uh at a point x= c only if f ofx is zero undefined or discontinuous.

[01:25:27]I mean if it's undefined isn't it isn't that captured by discontinuous I don't know why it why they repeated this at x= c like only if the function value is zero or discontinuous I think is enough like if it's undefined then it's discontinuous the graph that follows at the left shows a function that is continuous on a to b and changes sign only at its roots.

[01:25:59]graph at the right is discontinuous somewhere and therefore can change sign as we move left to right without ever crossing the x the y axis yeah okay so two examples of functions The first function has a value here. [snorts] It has a root here. It has another root here.

[01:26:45]And then another root here. [snorts] Um, oops. Oh, we're getting Notability is trying to help me out here. I don't want that. Thank you, though.

[01:27:12]Okay.

[01:27:16]So in this case um this spot is a this spot is c1 this x value is c2 this x value is c3 and then this x value is b and f is continuous on the closed interval a to b and it changes sign at each of those zeros. Like anytime that the function changes sign, it it must be zero, undefined or discontinuous at that point.

[01:28:05]Uh this next example, we have a function with a discontinuity.

[01:28:24]And it changed sign even though the function value was not zero.

[01:28:54]Make math great again. Yeah. [laughter] I don't like I don't like it when we make things great again, but I Yeah, we should make math great. Yeah.

[01:29:06]Reinvent math. You won't. Of course not.

[01:29:09]Yeah. Unify all of math. No, I can't do that. Math sucks. Why would you say that, dude? Math math rocks, bro. Hey, uh, Santhos, thank you. Thank you. I'm glad that you're enjoying the basic math series. I I'm very proud of it and I'm I'm looking forward to seeing where where we go with it because I think it's got a lot of potential. I clearly there's a lot of people out there who, you know, are excited to be studying math for fun. I think math is awesome.

[01:29:42]Like it can be a good hobby that's good for your brain and just fun. I definitely definitely believe that I'm on video games while on the stream.

[01:29:54]Come on, dude. Come on, dude.

[01:30:00]Math can be your video game, dude. No, it doesn't disqualify you. No. No, bro.

[01:30:05]It sucks at some point always. Nah. I mean, there's different types of math. I don't really like analysis.

[01:30:13]If I'm going to be honest with you all, I don't love calculus. It's okay. Like, I I got decently good at it, but my favorite kind of math is more algebraic.

[01:30:23]I prefer the algebraic side of math. So, hey, uh, Jack Mac. I mean, if you're enjoying this, I'd say hang out here and just get a vibe for what we're doing.

[01:30:41]Like, over the next week, I'm going to be studying some complicated math, some like graduate level stuff that you're not going to follow unless you're a math major. Um, and that's okay. Like, maybe you still just get a vibe. Maybe you just get a sense. And hopefully some people I know some people like to just study along with me. like you study whatever it is that you're working on and hopefully some people can be productive, you know, studying together.

[01:31:15]So I I think yes, you can try. It works for some people, not for everybody though. Um, no people Siraj, if somebody's a channel member, their name is green. So, right now I don't think that we have any channel members at the moment in this chat. And that's fine. Like, dude, I do not want to be pressuring anybody. I'm not suggesting, you know, there's no expectation in my mind that anybody signs up to be a channel member. Like, it's just it's an honor to be able to hang out and practice math and share it for free with you all live. It's just I don't want like every single stupid day where I'm making random jokes to be permanently available forever and like analyzed and you know every dumb little arithmetic mistake that I've ever made. I don't know. I want like a little bit of privacy about my own mistakes and I don't know.

[01:32:18]Um mean value theorem. We haven't hit mean value theorem yet, bro. D. Martin, D. Martin, thank you, dude. If you're here, I appreciate you very much. Let me know. I would be happy. Yo, we got a hobbyist, too. D. Martin. Bro, we got a new hobbyist in the in the chat. Thank you, dude. I think I owe a few new names, by the way. D. Martin.

[01:32:48]D. Martin SCA.

[01:32:52]Dude, thank you so much. Um, no, I'm not done with the Lang book.

[01:32:58]We'll be doing it for a while. The great Yaya.

[01:33:03]Um, yeah, dude. Appreciate you all.

[01:33:06]Like, this makes me so happy and encouraged that that there are a bunch of math nerds out there, you know, who like what we're doing. So, I'm going to write down a name every single time somebody super chats like special people who have helped like Leo Lexus has not paid me a single dollar, but he's like a good friend of the channel and we've collaborated together. And you know, likewise with like Lock and Levi and Happy Abe and others, although Happy Abe just signed up, but yeah, these are all this is the thanks page. It's kind of like Patreon walls like YouTube creators will do that, you know. I want to do the same thing. Okay. Yo, De Mari, been enjoying the lesson so far. I've always loved math and you love the idea of studying it as a hobby. Keep it up. Hell yeah, dude. Since you're here, is there a certain type of problem or math topic that you'd like me to to work? Like for hobbyists, I will pause whatever I'm practicing and I can work a problem that you want to see. So, if you want, I could I could work something from the basic math book or I could work a problem from calculus or something else.

[01:34:21]Um, dude, appreciate you very much.

[01:34:23]Thank you. Um, and yeah, definitely check out the the VOD of the membersonly live stream from a couple days ago. Dude, thank you. Um, least favorite math for more people. All good. You're just hanging out. All right. Hell yeah, dude. Well, thanks.

[01:34:42]I'm still going to credit you then.

[01:34:46]I'm going to say that you are a sponsor of calculus today. Thank you, bro. D Martin SCA, you are now permanently tattooed in the exercise set.

[01:35:09]You're now permanently tattooed in the exercise set for continuity, bro. Thank you, Luke. Uh, huge and D Martin. Thank you for supporting continuity.

[01:35:24]We're almost done with this problem zero. I'm almost done taking notes.

[01:35:33]I'm basically like a hot tub streamer.

[01:35:35]I'll write your names for a dollar.

[01:35:39]Uh function can only change signs at roots and discontinuities.

[01:35:45]Now we have a few examples.

[01:35:49]Examples, examples, examples. Bunch of examples. I feel pretty good. I can skim these, but I I think that Oh, a real world illustration of extreme in Yeah.

[01:36:07]Yeah. Yeah, I'm good on these examples.

[01:36:09]I'll pass on the examples. We did it. We did it. Hell yeah. Problem zero complete.

[01:36:28]Lol. Yeah. Yeah. Hope you don't mind me tattooing your name in my books.

[01:36:34]Um, all right. Next problem.

[01:36:37]1.4 skills.

[01:36:41]We have a skills problem and this one says for each describe the intervals on which f is continuous.

[01:36:51]Um describe the intervals for which f is continuous.

[01:37:07]Uh, for each discontinuity, describe the type of discontinuity in any one-sided continuity.

[01:37:16]Justify your answers about discontinuities with limit statements.

[01:37:21]Okay.

[01:37:23]Um, I'm just going to give a dot dot dot on that.

[01:37:27]Said it out loud. Uh we have one, two, three, four ticks on the y axis and we have four ticks one 2 3 4 on the x- axis as well.

[01:37:42]Um we have a discontinuity at the point at x= 2.

[01:37:53]the function value is three and then otherwise um it looks like we have a line going up with a steep slope here and Yeah, this book does not draw arrows on the on the graph, but it looks something like this. And I assume that the book means that this graph continues in both directions. It's like other other books will use like a little arrow sign on the on the graph itself.

[01:38:54]Um I think that this book is trying to say that this function f is continuous on negative infinity to pos2 and positive 2 to infinity.

[01:39:25]So it is continuous on those intervals. These intervals do not have closed ends.

[01:39:34]So it the function is continuous on those intervals.

[01:39:41]It is discontinuous at um at x= 2.

[01:39:56]As the limit as x tends to 2 um equals one which does not equal the function value at two.

[01:40:17]The function value at two is three.

[01:40:21]Oh, shoot.

[01:40:25]Limit as x tends to 2 of f ofx is not equal to f of two. So, it is a uh removable discontinuity as patching or changing f with if I change the function value to be one um then this makes f continuous on its entire domain which I assume this book is saying is the real numbers seems like it gain a lot more value out of this than a hot tub streamer though. Yeah. I Well, I don't know, man. I mean, depends.

[01:41:40]Depends for some people, you know. Some people just I'm just saying some people might prefer to watch a hot tub streamer, and I'm okay with that. I understand. Is Thomas Calculus okay? Uh, I think it's okay. I like this book a lot better, though. This book is definitely a lot better in my opinion because it's more conceptual.

[01:41:59]This book is way more conceptual I think than Thomas.

[01:42:07]Yeah. Um no the the intuitive definition if I come back to the intuitive definition that people often say you'll see a lot of people say that a function is continuous if you can draw it the graph without picking up your pencil.

[01:42:29]Um so this type of graph is the type of failure that the exercise is. It has a discontinuity. It has a different um value somewhere than its limit value.

[01:42:47]Um so this is something called a removable discontinuity and but formally like not the intuitive definition you need you define continuity of a function at a point the left limit equals the right limit equals the function value.

[01:43:10]Um and then then you can define left continuous and right continuous as well. You define continuity on an interval.

[01:43:24]Um yeah and then you can classify discontinuity is some of the simple ones. So it's continuous on um those intervals. It's left continuous I suppose. goes.

[01:44:09]Okay. Yeah.

[01:44:15]Yeah, I I think I got this. Describe the intervals for which it's continuous.

[01:44:21]Um I think I have enough [snorts] here. I I showed that the limit of f ofx is one from either direction.

[01:44:39]uh the limit value is one, but it doesn't equal the function value.

[01:44:46]And it's a removable discontinuity because you can change that one function value and then it fixes it. Uh and then that fixes it.

[01:45:01]Um yeah, we're done. That's it.

[01:45:12]It's number 23.

[01:45:17]I worked Stewart calculus and I know Steuart and Thomas are are really really similar. They're very similar books. Um I've tutored out of Stuart a lot. That's the book that my university used for, you know, for the standards calculus sequence. This book is newer and I think it's really good. I think it's really good. Actually, I I really think Tolman is is really nice. It's both conceptual and um you know it's got both conceptual and applied con you know calculational problems as well.

[01:46:02]I like Tolman um applications. Speaking of applications, how are we on time? We're almost two hours in. Um, I should probably wrap this up soon. So, this will be the last problem and then, you know, farewell, good riddance, calculus. We'll end with an application problem as is appropriate for calculus.

[01:46:37]applications. Um, explain in practice here. You know what?

[01:46:42]I'm not going to bother writing this whole thing. It's a lot of words. I like writing the problems when I can, but there's a lot of words in this one.

[01:46:59]So we are looking at an application of the extreme value theorem.

[01:47:12]Explain in practical terms what the extreme value theorem says about each continuous function. Then explain in practical terms what the intermediate value theorem says in each situation. So solution Alina hasn't cut her hair for six years.

[01:47:32]Six years ago, her hair was just 2 in long. Now, her hair is 42 in long. Let h of t be the function that describes the length in inches of Alina's hair t years after she stopped cutting it. Okay.

[01:47:50]Um, h of t is defined on the interval um on the closed interval AB equal um 0 years since she stopped cutting it to today, which is 6 years after.

[01:48:29]Um by the extreme value theorem, uh there is for the minimum and maximum values of H.

[01:49:12]There is some point does the do they use M and capital M as the X value or the the function values?

[01:49:34]I want to get the same notation.

[01:49:39]Oh yeah. Okay.

[01:49:45]Let me let me fix this. Okay. So, by the extreme value theorem, there are points lowercase M in AB and capital M in AB.

[01:50:11]such that um If we assume that hair grows monotonically, monotonically, which means just strictly growing.

[01:51:12]Then lowerase M is zero and capital M is 6 years with F of lowerase m= 2 in and f of [snorts] capital n the maximum value is 42 in.

[01:51:48]Um that seems like a reasonable assumption. I don't think the hair grows and then shrinks and then grows again or anything like that. Um, yeah.

[01:52:03]Hey, Jorion, you came at the end. Yeah, I'm almost done for today. Um, what's up? It's an album image from one of Wang Lines's projects. Interesting.

[01:52:16]Con value theorem. Yeah, HFT should be monotonically increasing. Yes, it would be, I guess. Yeah, monotonically increasing.

[01:52:27]Your textbook needs to define like strictly monoton does monotonically increasing means strictly monotonically or because often you you say oh wait no no no.

[01:52:43]You say that if f of a is less than or equal to f of b uh for all a less than b, then often you call this function increasing, which sounds a little funny, but then if f of a is strictly less than f of b for all a less than b, then we call that I think monotonically ally increasing um if I remember correctly that's the interval h of 0 would be 2 and h of six would be 42 did I get that wrong yeah no I got that uh so yeah f I forgot that we are using h In other words, h of 0 is 2 and h of six is 42. That these are the um extreme values.

[01:54:02]Do we have to sketch a graph or anything like that? Um, I think hair growth is sublinear.

[01:54:14]I think that you would expect something like this or something for hair growth.

[01:54:21]So if this person had 2-in long hair, you know, maybe this is 2 in and now this isn't drawn to scale, but this is now 42 in.

[01:54:38]Then your closed interval is from 0 to 6.

[01:54:53]Couldn't there be a hair limit where it practically stops growing?

[01:54:58]Yeah, I mean it's a good question. I that's why I'm saying like I think that hair will continue growing but it will slow down like yeah I do think I do think that it will slow down at some point. Y is greater than zero. Of course, John's hair would be a parabola.

[01:55:18]Okay. Yeah. Why is So, I I tried to draw that y started at two, right? If the function value starts at two, then this would I think represent I'm using an iPad with Notability.

[01:55:38]iPad, Notability, Apple Pencil, AirPlay, Air Server, OBS, um all those things. All right. So now I think that that explains what um the extreme value theorem tells us about this continuous function. Practical terms, what does the intermediate value theorem say in each situation? Um by intermediate value theorem there is some point um let's call it C or X knot in A to B.

[01:56:33]Let's let's reword this. By intermediate value theorem for any hair length um let's call it L for length in the function values from f0 to f uh six.

[01:57:11]I think it's a little bit I shouldn't I shouldn't write it like this. I mean the that assumes that we know that it's monotonically increasing but um for any hair length between 2 and 42 uh there is some time t in 0 to fix such that h of t equals l.

[01:57:55]For example, if um what's her name again?

[01:58:11]If Alina um wants to cut her hair when it is exactly when it is um 40 in or 30 in or something.

[01:58:46]There is there was some time there is some t some number of years between uh 0 and six when she should have gotten a dang haircut.

[01:59:15]if she likes to cut her hair when it's 30 in. There was some point of time when she officially became lazy and was overdue for a haircut.

[01:59:27]I I think that that's like um that's like a translation of what intermediate value theorem is saying.

[01:59:35]There was some point of time where she became overdue for a haircut if her if her preference was 30 in. Uh centimeters are better than inches. Ah, according to you.

[02:00:12]I think that's pretty good. I think that counts.

[02:00:18]And fairly tired. I think I'm done for today. We've gone for two hours working some calculus problems. I really had fun today. As always, uh please do like and subscribe. And hey, if anybody wants to share the basic if she cuts her hair at 30 in, well, she I'm saying that there was some time when it was 30 in. That's that's what the intermediate value theorem says. It's not like her hair suddenly overnight went from looking good at 25 in and then suddenly it became 42. right? It didn't just jump overnight and become too hairy. Uh there was some point in time when she should have gotten a haircut and now she's overdue for that. According to 95% of the planet rather, isn't this the isn't Yeah, this is the intermediate value theorem that we're talking about the second check mark here. So, let me bring up the definition again.

[02:01:26]If f is continuous on the interval for any function value between um the two end points there exists at least one point in the interior such that f takes that function value.

[02:01:46]So there exists some point um if I take K to be 30 then there exists a time such that f of t equal 30. There was a time when Alina had perfect hair at like 30 in perfect hair.

[02:02:25]Yeah. Assuming that they grow continuously and they started at under.

[02:02:29]Yes. Yeah. Those are the assumptions in the problem.

[02:02:35]It says continuous function assume that assume that this is a continuous function that hair growth is continuous. I think that's reasonable assumption like approximately um and that the problem tells us that h of 0 is 2 and h of six is 42.

[02:03:10]So there is some time when Alina had perfect hair like and she should have gotten some photos done or something like if Alina's favorite hair length is 30 in then there is some point when when she had perfect length hair.

[02:03:32]There exists some time between zero and six years when she had perfect hair.

[02:03:39]Yeah. Yeah. And of course, if she cuts her hair, that's a discontinuity. That would be a jump discontinuity.

[02:03:46]The graph would probably look something like this. And then again, like this. And then again, like, you know, this is the lifetime of haircuts.

[02:03:58]Speaking of, I should get a haircut.

[02:04:13]something like this.

[02:04:17]Okay.

[02:04:18]Um and then yeah, certainly on that day she begins a new these are all jump discontinuities.

[02:04:40]Um, okay. Uh, so my my analogy that I was trying to make is that Alina wants to cut her hair when it is I'm just like adding some extra ideas as just one example of how you can describe what the intermediate value theorem says about this situation.

[02:05:17]Um, like for example, if a if Alina wants to cut her hair when it is 30 in, there is some time tally greater than zero and strictly less than six such that um her hair length was 30 and so she should have cut it then that's like the perfect hair length Maybe.

[02:06:16]Yeah, I can I can show the problem statement again right here.

[02:06:33]Oh, yeah. Yeah. We're starting to get to some barber's paradox stuff here. Like what if Alina cuts her own hair and she I don't know. Yeah. Barber's paradox.

[02:06:44]Yeah, exactly. That's what I'm trying to say. Or another saying that's probably more realistic is like her perfect hair length, her ideal hair length like that she loves is 30 in. Right? There was some day between zero and six years when she had her favorite hair length. There was some time, there was some day that was like the ideal perfect hair day.

[02:07:17]Yeah.

[02:07:19]The sentence is false. Tell me whether the sentence is true or not. Yeah. Yeah.

[02:07:23]There's all sorts of Parad paradoxical stuff there.

[02:07:42][snorts] Awesome. Thank you everybody.

[02:07:52]Well, thank you. I had a blast today.

[02:07:56]those W hair the days. That's why you got confused. Okay. You were claiming that at some point she had 30 inches of hair. Yes. Well, that is true. Oh, you thought that she immediately cut it at 30. Oh, okay. No, no, no. The the problem is saying that today her hair is 42 in and a long time ago her hair was 2 in long and she hasn't cut it between.

[02:08:20]So there was some day when her hair was exactly 30 inches or 30.1 inches or anything in any number in between if we assume this is a continuous function. Um, yeah. So, there was some day when her hair was 30 inches long is is really the only thing you you can say on this. And then I'm adding a little bit of extra imagination that that's her favorite hair length or something. Yeah. W hair days. [laughter] There was some day there was some W hair day within the last six years.

[02:09:00]Thanks everyone for watching. I had fun today. Um, if anybody wants to share what I'm doing with anybody else, uh, you can say this out loud.

[02:09:09]Mathanbefun.com.

[02:09:12]Oh, wait, no. Math can be a hobby.

[02:09:15]That's what I meant to say. Math can be a hobby.com playlist for basic math uh, videos.

[02:09:28]So, if you ever want to share what I'm doing with folks, that can be like a really useful um URL to share this with anybody. John's hair as a function has a negative derivative at t equal. Yeah.

[02:09:42]Yeah. My hair is shrinking because I'm an unc, dude. Yeah. We're not even considering hair loss and things like that. Like, what? Dude, I'm getting old, bro. So, I'm getting old. I'm an unc as the kids say. We didn't even have that word back in my day. Um, all right. Thanks for watching everybody. Uh, please do like and subscribe. And if you know anybody who likes math, uh, share it with them.

[02:10:13]It's, you know, live streams become 10 times more fun, you know, when you know people in the chat. So, invite your friends and let's all just geek out about math together. Uh, thanks.

[02:10:25]Appreciate you all. Take care. Bye bye.