Real Analysis 52 | Riemann Integral - Examples

[@douglasstrother6584]

My Calculus I & II Professor (Tony Tromba, UC Santa Cruz, Fall 1981) dropped the Dirichlet Function on us at the end of a Friday lecture to give something to snack on during Happy Hour.

[@Leslie.Green_CEng_MIEE]

Concerning the Riemann integral of the Dirichlet function, at 3:43 we have *“the rational points lie dense in the real number line”* (although this expression is not defined). Then at 4:25 we have *_“for any segment you choose on the real number line, you always find a rational number”._* How are we to understand the above statement, when we also have the statements: (1) the rational numbers are “countably infinite” (2) the real numbers are “uncountably infinite", given that an uncountable infinity is a much larger infinity than a countable infinity?

[@wtt274]

Excellent explanation ! Thank you sir .

[@mathswithbuka]

Awesome Explanation!!!! Now i understand what my lecturer has been saying. Thank you!

[@brightsideofmaths]

Thank you very much :)

[@jecodedoncjesuis875]

Thank you for what you are doing.

[@brightsideofmaths]

My pleasure! :)

[@mr.petersen7150]

I'm taking a foundations of analysis course and I'm struggling to understand the content, so in searching I found your videos "among others, but yours being particularly enlightening". With that said, do you have a preferred textbook for learning real analysis that you could recommend? Your videos are helping me succeed as a better person, I honestly cant thank you enough Sir!

[@brightsideofmaths]

I don't have a preferred text book but I really like Introductory Real Analysis by A. N. Kolmogorov; S.V. Fomin.

[@evionlast]

This is almost the end in a real analysis course, what's the direction next? I have enjoyed this series very much rewatched a few episodes to remind me a few important points.

[@brightsideofmaths]

There is still a lot to cover about the Riemann integral. However, after this, my Real Analysis course ends. Other (related) topics will be discussed in another series :)

[@evionlast]

@@brightsideofmaths it's been an absolute Joy I'll continue with complex analysis, another outstanding series, recommended.

[@brightsideofmaths]

@@evionlast Thanks :)

[@douglasstrother6584]

Final Exam Extra Credit Problem: Plot the Dirichlet Function. More points = more points!

[@megavarshinim4157]

can u do multivariable calculus ,partial Derivatives and Tangent Plane also plz

[@mastershooter64]

first :) will you cover the riemann stieljes integral? (not sure if that was the correct spelling for steljes)

[@brightsideofmaths]

I think that the Lebesgue-Stieltjes integral is more useful and I already explained this in my Measure Theory series :)

[@ariuwu1234]

Hello, what software do you use for these videos?

[@brightsideofmaths]

The free and perfect program Xournal :)

[@ariuwu1234]

@@brightsideofmaths thank you very much for your response!