Real Analysis | Riemann Integrability

We give the definition of the lower integral, upper integral, and what it means for a function to be integrable. A few results are also proven including the fact that any continuous function is Riemann integrable.
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[@jayd.5296]

Your real analysis videos have helped me understand higher level math concepts better than any other textbook, lecture, etc. Thank you for explaining these concepts so well!

[@axelperezmachado3500]

Great! I was thinking about refreshing all this stuff yesterday and here it is!

[@JBall7]

Best math channel on YT hands down

[@tdchayes]

I started an analysis course in college, but decided it was too detailed (I did Computer Science). Now that I'm tutoring High School students in Calculus I'm finding this a useful addition to my knowledge.. Tutoring (teaching) has firmed up my understanding of the usual calculus program, but this adds one more level of depth to the understanding.

[@elhoplita69]

I found this video to be so clear, well job and than you so much!!

[@alirezaghadami2929]

these kinda videos are great! keep it up. you re super

[@_seniru_dilmith_]

Thank you so much. This helped me a lot😊

[@kushagrasinghal8209]

you da goat man this was awesome

[@edrinhasaj6844]

thank you so much i have an exam in two days and this makes things so clear

[@rafael7696]

Great. I miss a video of lebesgue integral. Would be very interesting.

[@alieser7770]

Sir you are awesome

[@jimmyjonestodd2556]

Great shirt

[@kavindushehan8984]

Sir could you please make a video about theorem " if set of discontinuity points has a finite number of limit points then f is Riemann integrable " . Thanks in advance!!

[@blinkora1898]

Thank you so much 😘😘😘😘😘😘😘😘 you are the best please make us videos about solving functional equations

[@MichaelPennMath]

I have a few already, but I will plan to make a few more in the upcoming weeks.

[@ntokozomoumakwe1512]

CAN i PLEASE HAVE THE KINK @@MichaelPennMath

[@AayushDuttaswag]

Quick question: Isn’t this definition of an integral closer to that of the Darboux integral (which is equivalent to the traditional Riemann integral)?

[@MichaelPennMath]

You are right! The reference I am using for these videos (in the description) is using this as the definition of the Riemann integral. I could say that I am staying consistent for students in my class -- but I also didn't think to point out that this is traditionally the Darboux integral.

[@kalelhenrik5699]

@Keagan Malachi Definitely, I've been watching on InstaFlixxer for months myself :)

[@thomasjefferson6225]

​@@MichaelPennMath it's one of the few times the French and the Germans have agreed on something.

[@nathanisbored]

are we gonna learn a precise definition of a differential, in such a way that we will eventually be allowed to multiply and divide them like numbers (for use in solving DEs)? or will they still be relegated to being notational artifacts?

[@alphalunamare]

You might need Lebesgue for that?

[@djvalentedochp]

great

[@riadsouissi]

Hi Michael. At 2:58, you assume you can find L(f, P2) between U(f, P1) and L(f). And similarly, at 9:00, you assume you can find U(f, P1) between U(f) and U(f)+epsilon/2. Shouldn't you demonstrate first that U(f, P) and L(f, P) are continuous?

[@stashdsouza4602]

Script P contains partitions, which are more like objects than Real Numbers. If we could not find L (f, P2) > U (f, P1), then U (f, P1) would be an upper bound for all the lower sums, but since U (f, P1) < sup L this would be a contradiction. It would mean sup L is not the least upper bound as we found an upper bound smaller than sup L.

[@sainandanpanigrahy4715]

According to the definition of uniform continuity, isnt it supposed to be If(x) - f(y)I < epsilon ? Why epsilon/b-a at 13:16?

[@TavartDukod]

The deffinition of uniform continuity says this inequality is true for all epsilon. Notice, the name of the variable here doesn't matter, we can replace it with any other letter, for example rho. Than we get that for all positive rho there is such an delta that if |x - y| < delta than |f(x) - f(y)| < rho. Now we can just take rho = epsilon / (b - a). The confusing part is the criterion for integribility and the definition of uniform continuity both use the same name of a variable, but when used together is important to remember those are actually to different variables.

[@Tekashiixine-yb5kh]

the short hand you used for definition as def^n_ reminds me of the shorthand my precalc teacher used. Where is the shorthand gotten from?

[@MichaelPennMath]

I am not sure where I first saw it. I think it is pretty universal. I also use sol^n, f^n (for function), poly'al (for polynomial)

[@quantabot1165]

92K

[@kushagrasinghal8209]

could someone help me understand what happened at 2:30 I get the fist part of the inequality but dont get what forces U9(f,p) to be less than L(f)

[@kushagrasinghal8209]

nevermind I was able to write out the inequality proof - it does follow from defintiion anf assumptio n

[@loeursovannra]

9:00 Hello, why U(f,P1)

[@henk7747]

Add any real number to an infimum and it is no longer a lower bound. In this case U(f) + e/2 is no longer a lower bound for the set {U(f,P)|P is in a partition}. If it's no longer a lower bound, we can an element in it that is smaller than U(f) + e/2.

[@goodplacetostop2973]

19:16 Hey! I’m curious today. I was wondering from where you’re watching this video. Anyway, I didnt have time to find a nice homework so nothing today lol.

[@natepolidoro4565]

PA, USA

[@3manthing]

Ljubljana, which is capital city of the country of Slovenia (a very small country located on the far north of Balkans region of Europe. Slovenia is neighbouring, west on Italy, south/south-east on Croatia, north on Austria, north-east on Hungary).

[@MichaelPennMath]

I have a Slovenia math Olympiad exam open in a tab right now waiting for me to pick a problem.

[@djvalentedochp]

Rio de Janeiro, Brazil. What about you Good Place To Stop?

[@R0M4ur0]

Italy...

[@kartikraturi9888]

Michael Penn- Richard Feynman of mathematics

[@tomkerruish2982]

With all due respect, John Conway was the Richard Feynman of mathematics.

[@omfgmx]

Hey, that's a nice tee you've got there :D

[@mattdavies7312]

Could someone explain to me why at 8:58 epsilon/2 is used?

[@nahuu4481]

e/2 + e/2 = e

[@Selenatorgirl544]

@@nahuu4481 but how can we now that we are going to end up with this solution?

[@nahuu4481]

@@Selenatorgirl544 Das ist nur eine Definition, du kannst epsilon genauso definieren, wie du es willst. In diesem Fall hast du "zwei Teile " und dann wurde diese Definition (e= e/2) verwendet. (German, sorry but please Translate)

[@Selenatorgirl544]

@@nahuu4481 hahah ich lebe in Deutschland! vielen Dank für deine Antwort (:

[@nahuu4481]

@@Selenatorgirl544 Ohjaa haha das war total unerwartet. Hast du es eigentlich aber verstanden?

[@ashutoshtiwari3129]

Hey, theres need to be a little correction, the U(f)

[@maburwanejohannes4647]

My brain: What kind of maths is this?😵😵

[@cristianv2850]

Same lol. Im only 15 years old and my brain just shuts down.

[@notsojharedtroll23]

As long as the curiosity is there, you'll get used to it. It's a good aproach to learn Math for it's own sake