Lecture 38 of 52MODULE 8: INTEGRATION
Integration
Prof. S.H. Kulkarni · IIT Madras
Credits and licence
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- Real Analysis, IIT Madras (at NPTEL)
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only.
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
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All the lectures
- Introduction
- Functions and Relations
- Finite and Infinite Sets
- Countable Sets
- Uncountable Sets, Cardinal Numbers
- Real Number System
- LUB Axiom
- Sequences of Real Numbers
- Sequences of Real Numbers - continued
- Sequences of Real Numbers - continued...
- Infinite Series of Real Numbers
- Series of nonnegative Real Numbers
- Conditional Convergence
- Metric Spaces: Definition and Examples
- Metric Spaces: Examples and Elementary Concepts
- Balls and Spheres
- Open Sets
- Closure Points, Limit Points and isolated Points
- Closed sets
- Sequences in Metric Spaces
- Completeness
- Baire Category Theorem
- Limit and Continuity of a Function defined on a Metric space
- Continuous Functions on a Metric Space
- Uniform Continuity
- Connectedness
- Connected Sets
- Compactness
- Compactness - Continued
- Characterizations of Compact Sets
- Continuous Functions on Compact Sets
- Types of Discontinuity
- Differentiation
- Mean Value Theorems
- Mean Value Theorems - Continued
- Taylor's Theorem
- Differentiation of Vector Valued Functions
- Integration
- Integrability
- Integrable Functions
- Integrable Functions - Continued
- Integration as a Limit of Sum
- Integration and Differentiation
- Integration of Vector Valued Functions
- More Theorems on Integrals
- Sequences and Series of Functions
- Uniform Convergence
- Uniform Convergence and Integration
- Uniform Convergence and Differentiation
- Construction of Everywhere Continuous Nowhere Differentiable Function
- Approximation of a Continuous Function by Polynomials: Weierstrass Theorem
- Equicontinuous family of Functions: Arzela - Ascoli Theorem