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Real analysis is the area of mathematics dealing with real numbers and the analytic properties of real-valued functions and sequences. The **[course_title]** is designed to develop concepts such as convergence, continuity, completeness, compactness, and convexity in the settings of real numbers, Euclidean spaces, and more general metric spaces. It shows the utility of abstract concepts and teaches an understanding and construction of proofs.

**Assessment**

This course does not involve any written exams. Students need to answer 5 assignment questions to complete the course, the answers will be in the form of written work in pdf or word. Students can write the answers in their own time. Each answer needs to be 200 words (1 Page). Once the answers are submitted, the tutor will check and assess the work.

**Certification**

Edukite courses are free to study. To successfully complete a course you must submit all the assignment of the course as part of the assessment. Upon successful completion of a course, you can choose to make your achievement formal by obtaining your Certificate at a cost of £49.

Having an Official Edukite Certification is a great way to celebrate and share your success. You can:

- Add the certificate to your CV or resume and brighten up your career
- Show it to prove your success

Course Credit: Open Culture

### Course Curriculum

Module: 01 | |||

Mod-01 Lec-1 Introduction | 00:53:00 | ||

Mod-01 Lec-02 Functions and Relations | 00:51:00 | ||

Mod-01 Lec-3 Finite and Infinite Sets | 00:51:00 | ||

Mod-01 Lec-4 Countable Sets | 00:50:00 | ||

Mod-01 Lec-5 Uncountable Sets, Cardinal Numbers | 00:50:00 | ||

Module: 02 | |||

Mod-02 Lec-06 Real Number System | 00:52:00 | ||

Mod-02 Lec-7 LUB Axiom | 00:52:00 | ||

Mod-02 Lec-08 Sequences of Real Numbers | 00:52:00 | ||

Mod-02 Lec-09 Sequences of Real Numbers – continued | 00:52:00 | ||

Mod-02 Lec-10 Sequences of Real Numbers – continued… | 00:51:00 | ||

Mod-02 Lec-11 Infinite Series of Real Numbers | 00:52:00 | ||

Mod-02 Lec-12 Series of nonnegative Real Numbers | 00:53:00 | ||

Mod-02 Lec-13 Conditional Convergence | 00:54:00 | ||

Module: 03 | |||

Mod-03 Lec-14 Metric Spaces: Definition and Examples | 00:53:00 | ||

Mod-03 Lec-15 Metric Spaces: Examples and Elementary Concepts | 00:56:00 | ||

Mod-03 Lec-16 Balls and Spheres | 00:52:00 | ||

Mod-03 Lec-17 Open Sets | 00:51:00 | ||

Mod-03 Lec-18 Closure Points, Limit Points and isolated Points | 00:52:00 | ||

Mod-03 Lec-19 Closed sets | 00:51:00 | ||

Module: 04 | |||

Mod-04 Lec-20 Sequences in Metric Spaces | 00:52:00 | ||

Mod-04 Lec-21 Completeness | 00:49:00 | ||

Mod-04 Lec-22 Baire Category Theorem | 00:53:00 | ||

Module: 05 | |||

Mod-05 Lec-23 Limit and Continuity of a Function defined on a Metric space | 00:53:00 | ||

Mod-05 Lec-24 Continuous Functions on a Metric Space | 00:54:00 | ||

Mod-05 Lec-25 Uniform Continuity | 00:51:00 | ||

Module: 06 | |||

Mod-06 Lec-26 Connectedness | 00:40:00 | ||

Mod-06 Lec-27 Connected Sets | 00:55:00 | ||

Mod-06 Lec-28 Compactness | 00:51:00 | ||

Mod-06 Lec-29 Compactness – Continued | 00:52:00 | ||

Mod-06 Lec-30 Characterizations of Compact Sets | 00:56:00 | ||

Mod-06 Lec-31 Continuous Functions on Compact Sets | 00:53:00 | ||

Mod-06 Lec-32 Types of Discontinuity | 00:55:00 | ||

Module: 07 | |||

Mod-07 Lec-33 Differentiation | 00:53:00 | ||

Mod-07 Lec-34 Mean Value Theorems | 00:50:00 | ||

Mod-07 Lec-35 Mean Value Theorems – Continued | 00:51:00 | ||

Mod-07 Lec-36 Taylor’s Theorem | 00:50:00 | ||

Mod-07 Lec-37 Differentiation of Vector Valued Functions | 00:51:00 | ||

Module: 08 | |||

Mod-08 Lec-38 Integration | 00:51:00 | ||

Mod-08 Lec-39 Integrability | 00:51:00 | ||

Mod-08 Lec-40 Integrable Functions | 00:51:00 | ||

Mod-08 Lec-41 Integrable Functions – Continued | 00:51:00 | ||

Mod-08 Lec-42 Integration as a Limit of Sum | 00:52:00 | ||

Mod-08 Lec-43 Integration and Differentiation | 00:52:00 | ||

Mod-08 Lec-44 Integration of Vector Valued Functions | 00:52:00 | ||

Mod-08 Lec-45 More Theorems on Integrals | 00:52:00 | ||

Module: 09 | |||

Mod-09 Lec-46 Sequences and Series of Functions | 00:51:00 | ||

Mod-09 Lec-47 Uniform Convergence | 00:52:00 | ||

Mod-09 Lec-48 Uniform Convergence and Integration | 00:53:00 | ||

Mod-09 Lec-49 Uniform Convergence and Differentiation | 00:52:00 | ||

Mod-09 Lec-50 Construction of Everywhere Continuous Nowhere Differentiable Function | 00:52:00 | ||

Mod-09 Lec-51 Approximation of a Continuous Function by Polynomials: Weierstrass Theorem | 00:51:00 | ||

Mod-09 Lec52 Equicontinuous family of Functions: Arzela – Ascoli Theorem | 00:53:00 | ||

Assessment | |||

Submit Your Assignment | 00:00:00 | ||

Certification | 00:00:00 |

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