The problem from the proof outline from 11/28 covered the definition of a nondecreasing function. We learned about the nondecreasing sequences but not nondecreasing functions in this course. In this post, I would like to review the definition of a non-decreasing function and further explore relative definitions of a nonincreasing function, an increasing function, a […]

In this class, the topic of Riemann Integration was introduced by Dr. K. This post will cover the definitions and lemmas that were covered during class. Note that the definitions build on another because we slowly built up to the idea of being Riemann Integrable. 1. A function f is bounded on [a,b] if is […]

In this definitions blog, I will explore the supremum norm in greater detail. First, recall the definition of the supremum norm: If , then we set the supremum norm . for the “uniform size” of and is the “uniform distance” between and . Notice that the word “uniform” comes up in this definition. Recall previous […]

In this post we will be looking at uniform continuity. To investigate the concept, let’s first look at what we already know. “Uniform” has shown up previously in the definition of uniform continuity. Comparing it to plain continuity will be insightful. Notice, then, how pointwise continuous functions have the sole requirement that we can meet […]

In class we discussed Darboux’s Theorem and the oddity that it implies. Differentiability of a function does not imply the continuity of its derivative, but it does imply the Intermediate Value Theorem holds for its derivative. For this blog, I would like to explore a function for which this oddity occurs. Consider the function discussed […]

The Intermediate Value Theorem: In the book, we have the formal definition stating: Let f : [a, b] → R be continuous. If L is a real number satisfying f(a) < L < f(b) or f(a) > L > f(b), then there exists a point c ∈ (a, b) where f(c) = L. To prove […]

During the lecture on Tuesday we continued our discussion of uniform continuity by discussing Theorem 4.4.7. This theorem seems as if it will be very useful in the future and thus I would like to dig deeper into it. 1. Preliminaries First let us recall definitions and theorems that will be mentioned. Compact Set: a […]

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 […]

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