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# 数学代写|数学分析作业代写Mathematical Analysis代考|Continuity and Equivalent Metrics

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## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Continuity and Equivalent Metrics

Continuity, from the intuitive point of view, is about the gradual rather than the abrupt change of function values. In its simplest form, the graph of a continuous, real-valued function of a single real variable must be connected. Most functions in mathematics are too complicated for such a visual characterization of continuity, and a more rigorous and robust definition is needed. The $\epsilon-\delta$ definition of continuity revolutionized calculus, and hence mathematics, in the early nineteenth century. It is based on the idea that the fluctuations of a continuous function can be controlled in a sufficiently small neighborhood of a point of continuity. Our definition of local continuity in the metric setting is an immediate generalization of the $\epsilon-\delta$ definition. We then define the global continuity of a function on a metric space, an important concept seldom treated in undergraduate textbooks. You will see that continuity does not depend on the specific metric we use to measure proximity, but rather on the collection of open sets the metric induces. This leads us to the notion of equivalent metrics and, more generally, homeomorphisms.
Definition. Let $(X, d)$ and $(Y, \rho)$ be metric spaces. A function $f: X \rightarrow Y$ is said to be continuous at a point $x_0 \in X$ if, for every $\epsilon>0$, there exists $\delta>0$ such that $\rho\left(f(x), f\left(x_0\right)\right)<\epsilon$ whenever $d\left(x, x_0\right)<\delta$.
The following theorem is an obvious restatement of the definition.
Theorem 4.3.1. Let $(X, d)$ and $(Y, \rho)$ be metric spaces. A function $f: X \rightarrow Y$ is continuous at $x_0$ if and only if the inverse image of an open ball in $Y$ centered at $f\left(x_0\right)$ contains an open ball in $X$ centered at $x_0$.

## 数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Homeomorphisms

The concept of a homeomorphism is of central importance in topology. In the metric setting, isometry, although quite useful, is too stringent and does not preclude homeomorphisms from being useful. One can loosely think of a homeomorphism as a relaxation of the concept of isometry and an extension of the notion of metric equivalence.
Definition. Two metric spaces $(X, d)$ and $(Y, \rho)$ are homeomorphic if there exists a bicontinuous bijection $\varphi$ from $X$ to $Y$. The function $\varphi$ is called a homeomorphism from $X$ to $Y$.
Example 10. The open interval $(-1,1)$ is homeomorphic to $\mathbb{R}$ (both sets have the usual metric). The function $f(t)=\frac{t}{1-t^2} \operatorname{maps}(-1,1)$ bicontinuously onto $\mathbb{R}$.
Example 11. The closed upper half plane $H=\mathbb{R} \times[0, \infty)$ is homeomorphic to the half-open strip $A=\mathbb{R} \times[0,1)$. To see this, define $\varphi: H \rightarrow A$ by $\varphi(x, y)=$ $\left(x, \frac{y}{1+y}\right)$. It is a rather routine matter to verify that $\varphi$ is a bijection and that its inverse is $\varphi^{-1}(x, t)=\left(x, \frac{t}{1-t}\right)$.
Example 12 (the stereographic projection). Let $\mathcal{S}^1=\left{\left(\xi_1, \xi_2\right) \in \mathbb{R}^2: \xi_1^2+\xi_2^2-\right.$ $\left.\xi_2=0\right}$ be a circle of diameter 1 and centered at the point $(0,1 / 2)$, and let $N=$ $(0,1)$ be the top point on the circle. Define the punctured circle to be the circle with the top point removed: $\mathcal{S}^1=\mathcal{S}^1-{N}$. We give $\mathcal{S}^1$ the Euclidean metric in the plane. Define a bijection $P: \mathcal{S}^1 \rightarrow \mathbb{R}$ as follows: for a point $\xi=\left(\xi_1, \xi_2\right) \in \mathcal{S}^1$, $P(\xi)$ is the horizontal intercept of the line that contains the points $N$ and $\xi$, as shown in figure 4.1.

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。