## 数学代写|实分析作业代写Real analysis代考|MATH2350

2022年12月26日

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## 数学代写|实分析作业代写Real analysis代考|H¨older and Lipschitz Continuity

Sometimes we deal with functions that are “better than continuous” yet are “not quite differentiable.” The next definition gives one way to quantify behavior that lies between continuity and differentiability.

Definition 1.4.1 (Hölder and Lipschitz Continuous Functions). Let $I$ be an interval in the real line, and let $f: I \rightarrow \mathbb{C}$ be a function on $I$.
(a) We say that $f$ is Hölder continuous on I with exponent $\alpha>0$ if there exists a constant $K \geq 0$ such that
$$|f(x)-f(y)| \leq K|x-y|^\alpha, \quad \text { for all } x, y \in I .$$
(b) If $f$ is Hölder continuous with exponent $\alpha=1$, then we say that $f$ is Lipschitz continuous on $I$, or simply that $f$ is Lipschitz. That is, $f$ is Lipschitz if there exists a constant $K \geq 0$ such that
$$|f(x)-f(y)| \leq K|x-y|, \quad \text { for all } x, y \in I .$$
A number $K$ for which this holds is called a Lipschitz constant for $f$.
By using the Mean Value Theorem, we can see that any function $f: I \rightarrow \mathbb{C}$ that is differentiable everywhere on $I$ and has a bounded derivative $f^{\prime}$ is Lipschitz on $I$ (this is Problem 1.4.2). However, a Lipschitz function need not be differentiable at every point. For example, $f(x)=|x|$ is Lipschitz on $[-1,1]$ but it is not differentiable at $x=0$.

Lipschitz functions will appear frequently in the text. In Chapter 5 we will prove that every Lipschitz function on $[a, b]$ has bounded variation and is absolutely continuous. We will encounter Hölder continuous functions with exponents $\alpha<1$ less frequently. The Cantor-Lebesgue function, which will be introduced in Section 5.1, is one important example of a Hölder continuous function that is not Lipschitz.

## 数学代写|实分析作业代写Real analysis代考|Lebesgue Measure

We know how to determine the volume of cubes, rectangles, spheres, and some other special subsets of $\mathbb{R}^d$. Does every subset of $\mathbb{R}^d$ have a volume? We are tempted to believe that each set $E \subseteq \mathbb{R}^d$ can be assigned a unique “volume” or “measure” $|E|$ in such a way that the following properties hold:
(i) $0 \leq|E| \leq \infty$
(ii) the measure of the unit cube $Q=[0,1]^d$ is $|Q|=1$,
(iii) if $E_1, E_2, \ldots$ are finitely or countably many disjoint subsets of $\mathbb{R}^d$, then
$$\left|\bigcup_k E_k\right|=\sum_k\left|E_k\right|,$$
(iv) $|E+h|=|E|$ for all $h \in \mathbb{R}^d$.
We will prove in Section $2.4$ that there is no way to define $|E|$ so that all four conditions (i)-(iv) simultaneously hold for every set $E \subseteq \mathbb{R}^d$ ! (This turns out to be a consequence of the Axiom of Choice; see Theorem 2.4.4.) Even so, we will prove in this chapter that if we relax our goal of defining a volume for every subset of $\mathbb{R}^d$, then we can create a useful definition of measure that satisfies properties (i)-(iv) for a very large class of subsets of $\mathbb{R}^d$. This class of “good sets,” which we will call the measurable subsets of $\mathbb{R}^d$, includes almost every set that we ever encounter in practice. The “volume” $|E|$ that we will define is called the Lebesgue measure of the set $E$; we will show that it is well-defined and “nicely behaved” on the class of measurable subsets of $\mathbb{R}^d$.
The creation of Lebesgue measure is a two-step process, broadly outlined as follows. First, we start with a basic class of subsets of $\mathbb{R}^d$ that we know how we want to measure. There are several choices for this class, but perhaps the simplest is the collection of rectangular boxes (rectangular parallelepipeds) in $\mathbb{R}^d$. The volume of a rectangular box is just the product of the lengths of its sides. We attempt to extend the notion of volume to arbitrary subsets of $\mathbb{R}^d$ by covering them with rectangular boxes in all possible ways. For each set $E \subseteq \mathbb{R}^d$, this gives us a number $|E|_e$ that we call the exterior Lebesgue measure of $E$. Every subset of $\mathbb{R}^d$ has a uniquely defined exterior measure, and the function $|\cdot|_e$ satisfies properties (i), (ii), and (iv) from our list above for every set $E$. However, there exist disjoint sets $A$ and $B$ in $\mathbb{R}^d$ such that $|A \cup B|_e<|A|_e+|B|_e$ ! Thus exterior Lebesgue measure does not satisfy property (iii) for all choices of disjoint subsets of $\mathbb{R}^d$.

# 实分析代写

## 数学代写|实分析作业代写Real analysis代考|H¨older and Lipschitz Continuity

(a) 我们说 $f$ Hölder 在 I 上连续吗? $\alpha>0$ 如果存在常数 $K \geq 0$ 这样
$|f(x)-f(y)| \leq K|x-y|^\alpha, \quad$ for all $x, y \in I$.
(b) 如果 $f$ Hölder 是指数连续的吗 $\alpha=1$ ，那么我们说 $f$ Lipschitz 连续吗 $I$ ，或者简单地说 $f$ 是利普希茨。那是， $f$ 如果存在常数，则为 Lipschitz $K \geq 0$ 这样 $|f(x)-f(y)| \leq K|x-y|, \quad$ for all $x, y \in I$

Lipschitz 函数会在文中频繁出现。在第 5 章中，我们将 证明每个 Lipschitz 函数在 $[a, b]$ 具有有限的变化并且是 绝对连续的。我们会遇到带指数的 Hölder 连续函数 $\alpha<1$ 不太频繁。将在 $5.1$ 节中介绍的 Cantor-
Lebesgue 函数是非 Lipschitz 的 Hölder 连续函数的一 个重要示例。

## 数学代写|实分析作业代写Real analysis代考|Lebesgue Measure

(i) $0 \leq|E| \leq \infty$
(ii) 单位立方体的量度 $Q=[0,1]^d$ 是 $|Q|=1$ ，
(iii) 如果 $E_1, E_2, \ldots$ 是的有限个或可数个不相交的子集 $\mathbb{R}^d$, 然后
$$\left|\bigcup_k E_k\right|=\sum_k\left|E_k\right|,$$
(四) $|E+h|=|E|$ 对所有人 $h \in \mathbb{R}^d$.

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## MATLAB代写

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