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 […]
The muddiest point for me from the lecture on Tuesday is visualizing a proposition we proved. The proposition states that if a function g from [0,1] to [0,1] is bijective and increasing, then the sum of its integral from 0 to 1 and its inverse’s integral from 0 to 1 is 1. It’s not essentially […]
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 […]
Upper and Lower Sums We illustrated lower sums in class to show that, given a function that is bounded on , where is a refinement of a given partition of . This post shows the analogous graph for upper sums that we did not draw in class. These are our observations about the areas in […]
For me, the muddiest point of this lecture was understanding the order of choosing and . I will illustrate this in the proof of Theorem 6.2.6. Statement of Proof: If uniformly on and continuous on for all , then is continuous on as well. Proof: Let . Choose such that . Choose such that implies . By the […]
Any point that warrants use of more than one analogy is probably a fairly “muddy” point. In class, we discussed the negation of uniform convergence using a cattle analogy. For uniform convergence the idea is, if each cow is a value of x, and marks the positions of the cows at time as they are […]
Without Loss of Generality In our proof of the Interior Extremum Theorem, we advanced our argument by supposing that in order to find a sign for so that the order limit theorem could be used. In doing this we put a restriction on the element in our proof that is not necessarily true for all […]
One of the things we did in the class was proving Theorem 4.4.7, the Uniform Continuity on Compact Sets. Theorem 4.4.7 states that a function is continuous on a compact set K is uniformly continuous on K. To me the prove in the class was hard to follow because the definition of compact set was […]
I think the muddiest point from Thursday’s class is the distinction between Uniform continuity and point-wise continuity. First, we recall the definitions of pointwise continuity and uniform continuity. Uniform continuity: A function f : A → R is uniformly continuous on A if for every ε > 0 there exists a δ > 0 such […]
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