Authors: Brittney D’Oleo, Elise Favia Date: October 27, 2017 Introduction: This week’s topic, (global) modulus of continuity is an extension of last week’s topic, the (local) modulus of continuity. In the blog, we will explore the new definition and compare it to last weeks definitions. We will also prove an interesting result creating an […]
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 […]
On Thursday 10/19 we went over uniform convergence and the Sequential Characterization, both of which I will address in this post. Definition: Given , (depending only on ) so that it holds that whenever , . In class we went over three functions, two of which I will touch on here. The first is the […]
Correction: Correction:
Consider the function . We will consider the domains and . A) To prove that is not uniformly continuous on , we need to show that for some , there exist sequences such that , but . Let and for . Notice that . Yet for all . Hence by the archimedean property we can choose . Thus is […]
Assume that is defined on an open interval and it is known to be uniformly continuous on and where . Prove that is uniformly continuous on . Assume is defined on an open interval . Suppose is uniformly continuous on and with . Suppose . Since g is uniformly continuous on , we […]
One of the things we did in this class was introduce and go through the proof for Theorem 4.2.3 (Sequential Criterion for Functional Limits). The theorem states: Given a function and , the following are equivalent: i.) ii.) sequences such that and it follows that For me, I think that the muddiest point lies in […]
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