In class we discussed the Sequential Criterion for Functional Limits, Divergence Criterion for functional limits, and Characterizations of continuity. We emphasized that we have built an understanding of functional limits and convergence that allow us to use several definitions interchangeably depending on the proof we are trying to complete. Theorem 4.2.3 (Sequential Criterion for Functional […]
By: Rhiannon Begley and Shuyi Chen Introduction In class we have begun to and will continue to discuss functions, their continuity, and their limits at a particular point. Although we can determine the continuity and limits of functions with the tools we have learned in class, we cannot determine how quickly the function is converging […]
Definition of Domain : Given the function we say that the domain of f is the set . As we discussed today in class the domain is not simply all the values that your function can take in but rather all the values that you decide your function will take in. Definition is the limit- […]
We began today’s class by going through the challenge problems. While we were talking about problem number 3, an interesting issue came up, which is how do we define the domain of a function. After our discussion, we found out that the domain of a function is not determined by the formula. Instead, the domain […]
During today’s class, we firstly addressed the challenge problems from chapter4. From the discussion of the challenges, I thought the muddiest point of today should be “how domain is define”. According to our common sense, for a specific function f(x), its domain should be the set of all possible values of x where f(x) is […]
By Nick Wan and Elaine Wissuchek Introduction to the Cantor Set Cantor set is a special subset of the closed interval [0, 1] invented by a German mathematician Georg Cantor in 1883. In order to construct this set, we need to construct infinitely many subset of inductively and take the intersection of all of them. […]
We began class today with a Weekly Situation. This situation was about what we can say about the set (a,b) from definitions versus what we can say using known theorems, lemmas, and homework results. We finished our discussion of Chapter 3 by proving part of the Heine-Borel Theorem. There are three statements in the Heine-Borel […]
I think the muddiest point from Thursday’s class is the distinction between sequences and sets that came from the opening situation. This muddiness most likely comes from our extensive discussions about sequences in the previous weeks, so we are still thinking in terms of sequences instead of now moving on to sets. Further confusion may […]
By: Jonathan Rodriguez and Shivani Patel Introduction: In this class, we often work with different parts of the real numbers, such as natural numbers, integers, and rational numbers. We can easily take real numbers for granted since they seem to encompass everything that we work with in the class. In order to fully appreciate the […]
Last Thursday we ended class with beginning to prove Theorem 2.6.4 which states that a sequence converges is Cauchy. We picked up on Tuesday with a bit of review of this proof: We are assuming is Cauchy and wanting to prove that converges. Using Lemma 2.6.3 and Bolzano – Weierstress we concluded that there exists […]
You must be logged in to post a comment.