# 微积分网课代修|积分学代写Integral Calculus代考|MTH-200 Full covers of a set

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## 微积分作业代写calclulus代考|Full covers of a set

A covering relation is simply a collection $\beta$ of interval-point pairs $([c, d], x)$ where $[c, d]$ is a compact interval and $x$ some point belonging to $[c, d]$.

DEFinition 2.1. A covering relation $\beta$ is full at a point $x_{0}$ if there is $\delta>0$ so that $\beta$ contains all pairs $\left([c, d], x_{0}\right)$ for which $c \leq x_{0} \leq d$ and $0<d-c<\delta$.

DEFinition 2.2. A covering relation $\beta$ is full cover of a set $E$ if $\beta$ is full at each point $x_{0}$ belonging to the set $E$.
2.4.1. Mandatory Exercises. At the risk of sounding too severe, we instruct the reader to master each of the following exercises. All of the calculus applications of these ideas are based on them.

ExERCISE 10 (Continuity at a point). Let $F$ be continuous at a point $x_{0}$, let $\epsilon>0$, and write
$$\beta=\left{\left([c, d], x_{0}\right): c \leq x_{0} \leq d \text { and }|F(d)-F(c)|<\epsilon\right} .$$ Show that $\beta$ is full at $x_{0}$. SOLUTION IN SECTION 8.2.1. EXERCISE 11 (Continuity at a point). A smaller and more useful covering relation uses the notion of oscillation ${ }^{1}$. Let $F$ be continuous at a point $x_{0}$, let $\epsilon>0$, and write
$$\beta=\left{\left([c, d], x_{0}\right): c \leq x_{0} \leq d \text { and } \omega F([c, d])<\epsilon\right} .$$ Show that $\beta$ is full at $x_{0}$. SOLUTION IN SECTION 8.2.2. EXERCISE 12 (Continuity at points in a set). Let $F$ be continuous at each point of a set $E$, let $\epsilon>0$, and write
$$\beta={([c, d], x): c \leq x \leq d \text { and }|F(d)-F(c)|<\epsilon} .$$
Show that $\beta$ is a full cover of $E$.
SOLUTION IN SECTION 8.2.3.

## 微积分作业代写calclulus代考|Full covers and Cousin covers

We prefer to say merely that $\beta$ is a full cover if $\beta$ is a full cover of $\mathbb{R}$, i.e., if $\beta$ is full at every real number. The entire theory of differentiation and integration of the calculus can be presented in a way that directly relates to full covers.

If a covering relation $\beta$ is a full cover then we have expressed the opinion that it should contain a partition of any interval $[a, b]$, i.e., there should be a subset $\pi$ of $\beta$,
$$\pi=\left{\left(\left[a_{i}, b_{i}\right], \xi_{i}\right): i=1,2, \ldots, n\right}$$
so that the intervals
$$\left{\left[a_{i}, b_{i}\right]: i=1,2, \ldots, n\right}$$
form a nonoverlapping collection of subintervals that make up all of $[a, b]$.
Note that, if our goal is to have partitions of $[a, b]$, we do not quite need $\beta$ to be full at the endpoints $a$ and $b$ since we would use only subintervals of $[a, b]$ and not concern ourselves with what is happening on the left at $a$ or what is happening on the right at $b$. This leads to the following definition, which slightly relaxes the condition defining full covers and also focusses on our need for partitions.

Definition 2.3. A covering relation $\beta$ is a Cousin cover of the compact interval $[a, b]$ provided that, at each point $x$ in $[a, b]$, there is a $\delta>0$ so that $\beta$ contains all pairs $([c, d], x)$ for which $c \leq x \leq d,[c, d] \subset[a, b]$ and $0<d-c<\delta$.

## 微积分作业代写calclulus代考|Full covers of a set

|beta $=\mid$ left $\left{\mid\right.$ left $\left([c, d], x_{-}{0} \mid\right.$ right): $c \mid$ leq $x_{-}{0} \backslash$ leq d $\mid$ text ${$ 和 $}|F(d)-F(c)|<\mid$ epsilon $\mid$ right $}$ 。 显示 $\beta$ 满了 $x_{0}$. 第 $8.2 .1$ 节中的解决方案。练习 11 (某一点的连续性) 。更小更有用的 覆盖关系使用振薃的概念 1 . 让 $F$ 在某一点上是连续的 $x_{0}$ ，让 $\epsilon>0$ ，和写
|beta= $=$ left $\left{\backslash\right.$ left $\left([c, d], x_{-}{0} \mid\right.$ right): $c \backslash$ leq $x_{-}{0} \backslash$ leq $d \backslash$ text ${$ and $} \backslash$ lomega $F([c, d])<\backslash \varepsilon \mid$ 对 $}$ 。 显示 $\beta$ 满了 $x_{0}$. 第 $8.2 .2$ 节中的解决方案。练习 12 (一组点的连续性) 。让 $F$ 在集合的 每个点上是连续的 $E$ ，让 $\epsilon>0$ ，和写
$$\beta=([c, d], x): c \leq x \leq d \text { and }|F(d)-F(c)|<\epsilon .$$ 显示 $\beta$ 是一个完整的封面 $E$. 第 8.2.3 节中的解决方案。