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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.


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.


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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
Submit Your Assignment 00:00:00
Certification 00:00:00

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