Also I think I have a great idea! It would be so nice that you would recommend a book for each playlists! Maybe by adding another video or mentioning a book or two in the intro!
Sorry I don’t know much about the rigorous math behind it, but I just wondered if these 2 things are related, and why/why not the expansion of the theorem cannot be applied this way
@@harrisongaga Short answer: no, the (common) derivative does not exist. Long answer: I have a whole video series about the extension in this direction :) See description and search for "Distributions" :)
This video reminded me of another question I had in the past: We all know that if a function is continuous then it has a primitive which can be given by the Fundamental Theorem of Calculus (FToC). However, some derivatives are not continuous and therefore it is not necessary for a function to be continuous to have a primitive. And this is a problem because we cannot rely on the FToC to prove the existence of this primitive as the theroem no longer holds. Now the challenge is: Find a necessary and sufficient condition for a function to have a primitive.
*Addendum:* Finding a necessary condition is easy, Draboux theorem states that derivatives must have the intermediate value property, this is a possible necessary condition, but not sufficient .. probably .. I honestly don't have any counterexamples at hand to prove this. *Further notes:* I had looked up this question on the internet and the answer I had found was beyond my comprehension and involved highly advanced stuff. I didn't check the proof as I knew it would be highly complicated I should check again the answer, I forgot how it looks like. I can't even recall whether I had found a proof.
![Fundamental Theorem of Calculus | Expansion of the Theorem [dark version]](http://i.ytimg.com/vi/_rH7hNrKxss/mqdefault.jpg)
![Fundamental Theorem of Calculus | Expansion of the Theorem [dark version]](/img/tr.png)
Also I think I have a great idea! It would be so nice that you would recommend a book for each playlists! Maybe by adding another video or mentioning a book or two in the intro!
Good idea!
Hi professor! Beautiful! In which playlist does this lesson fit?
Thanks! It's for the end of the Real Analysis playlist :)
For the step function (and maybe for the cantor function as well), isn’t it derivative the dirac delta function, which makes the integral 1
Sorry I don’t know much about the rigorous math behind it, but I just wondered if these 2 things are related, and why/why not the expansion of the theorem cannot be applied this way
@@harrisongaga Short answer: no, the (common) derivative does not exist. Long answer: I have a whole video series about the extension in this direction :) See description and search for "Distributions" :)
Wait for topology course.
Me too!
This video reminded me of another question I had in the past:
We all know that if a function is continuous then it has a primitive which can be given by the Fundamental Theorem of Calculus (FToC).
However, some derivatives are not continuous and therefore it is not necessary for a function to be continuous to have a primitive. And this is a problem because we cannot rely on the FToC to prove the existence of this primitive as the theroem no longer holds.
Now the challenge is: Find a necessary and sufficient condition for a function to have a primitive.
*Addendum:* Finding a necessary condition is easy, Draboux theorem states that derivatives must have the intermediate value property, this is a possible necessary condition, but not sufficient .. probably .. I honestly don't have any counterexamples at hand to prove this.
*Further notes:* I had looked up this question on the internet and the answer I had found was beyond my comprehension and involved highly advanced stuff. I didn't check the proof as I knew it would be highly complicated
I should check again the answer, I forgot how it looks like. I can't even recall whether I had found a proof.