## Lebesgue’s Criterion for Riemann Integrability

Weekly_Blog (1)

Weekly_Blog (1)

The last subject we covered in class was The Fundamental Theorem of Calculus which brings this course full circle with a proof that proves all the basic things we used in every Calculus class we have been in. I have stated the definition: Fundamental Theorem of Calculus: (i) If is integrable and then (ii) If […]

I think the muddiest point in today’s lecture was the relation between continuity, differentiability. The reason why I pick this point was that it is very easy to use the false conclusion when we are working on problems. In this post, I would start with the theorem in the textbook and then use graphs to provide […]

I thought that the muddiest point from last Thursday’s class was when we went over proving that . The reason why I thought that this was the muddiest is because the proof required to cases. A case in which k was negative and a case that case was positive . This is often a detail […]

Class of 11/30 basically talked about different Theorem related to the Properties of Integral. Those Theorem are built around the Fundamental Theorem of Calculus which we will discuss in next class meeting. We talked about Theorem 7.4.1 which states that Assume is bounded, and let . Then, f is integrable on [a, b] if and […]

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Tongzhou, 11/21 (a) Since needed to be determined firstly. By definition, . Since is any partition of , Therefore, . (b) Sidework: I want to have which means I want to determine how far could be away from . Thus so that . Now choose . Again I need to determine before I find . […]

In this edition of the muddiest point I will go over non-Riemann integrable functions. Let’s review a classic example that we mentioned in class, the Dirichlet function. Define the Dirichlet Function as $$ Recall that by definition, A bounded function defined on the interval is Riemann-integrable if . For , the set of all possible […]

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