## 核心能力 Competence

Ability to handle continuity phenomena and approximations.

Ability to operate with symbols and digits.

Ability to visualize space, shape, and images.

Ability of using computer to do mathematics.

Ability to apply stochastic mathematics.

Ability to handle large amount of data.

Ability to do mathematics relating to discrete models.

Ability to do independent studies of mathematics.

Abilities to use computer as a tool.

## 課程概述 Course Description

1.Real number system, Euclidean n-space 2.Point set topology: open and closed sets, boundary of a set, sequence and series 3.Compact and connected sets: Heine-Borel and Bolzano-Weierstrass theorems, path-connected and connected sets 4.Continuous mappings: boundedness of continuous functions on compact sets, intermediate-value theorem, uniform continuity 5.Differentiation: matrix representation, Taylor’s theorem, maxima and minima 6.Uniform convergence of sequence of functions: pointwise and uniform convergences, integration and differentiation of series, Arzela-Ascoli theorem, fixed point theorem, Stone-Weierstrass theorem, Dirichlet and Able test

## 課程學習目標 Course Objectives

• 建立學生基礎分析知識及技巧熟練
• 基本拓撲學及賦距空間之認識
• 高維度空間及抽象空間上微積分之應用
• ## 課程進度 Progress Description

進度說明 Progress Description
1Introduction of real number system
2Introduction of real number system
3Countability, completeness
4Sequence, Cluster points, Limit inferior and superior
5Normed vector space, inner product spaces and metric spaces
6Open set, closed set, boundary
7sequence and completeness
8Compactness, Connectedness
9Continuity, Operations on continuous maps
10Images of comapact sets, connected sets and path connected sets under continuous maps
11Uniform continuity
12Differentiation and Integration of functions of one variable
13Pointwise and uniform convergence
14Series of functions and the Weierstrass M-test
15Integration and Differentiation of series
16Space of continuous functions, Arzela-Ascoli theorem
17Stone-Weierstrass theorem, contraction mapping principle
18the existence and uniqueness of the solutions to ODEs
以上每週進度教師可依上課情況做適度調整。The schedule may be subject to change.

## 有關課程其他調查 Other Surveys of Courses

1.本課程是否規劃業界教師參與教學或演講?
Is there any industry specialist invited in this course? How many times?
2.本課程是否規劃含校外實習(並非參訪)?
Are there any internships involved in the course? How many hours?
3.本課程是否可歸認為學術倫理課程? 否
Is this course recognized as an academic ethics course? In the course how many hours are regarding academic ethics topics? No
4.本課程是否屬進入社區實踐課程? 否
Is this course recognized as a Community engagement and Service learning course? Which community will be engaged? No