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A Basic Course in Real Analysis

A Basic Course in Real Analysis. Instructor: Prof. P.D. Srivastava, Department of Mathematics, IIT Kharagpur. This is an introductory course in real analysis, covering topics: the Dedekind theory of irrational numbers; the Cantor theory of irrational numbers; bounded set and the open, closed and compact, etcetera; continuity of the function; differentiability; the Riemann integration and Riemann stieltjes integral; and the improper integral. (from nptel.ac.in)

Lecture 36 - Application of Mean Value Theorem, Darboux Theorem, L'Hospital's Rule

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Lecture 01 - Rational Numbers and Rational Cuts

Lecture 02 - Irrational Numbers, Dedekind's Theorem

Lecture 03 - Continuum and Exercises

Lecture 04 - Continuum and Exercises (cont.)

Lecture 05 - Cantor's Theory of Irrational Numbers

Lecture 06 - Cantor's Theory of Irrational Numbers (cont.)

Lecture 07 - Equivalence of Dedekind and Cantor's Theory

Lecture 08 - Finite, Infinite, Countable and Uncountable Sets of Real Numbers

Lecture 09 - Types of Sets with Examples, Metric Space

Lecture 10 - Various Properties of Open Set, Closure of a Set

Lecture 11 - Ordered Set, Least Upper Bound, Greatest Lower Bound of a Set

Lecture 12 - Compact Sets and its Properties

Lecture 13 - Weierstrass Theorem, Heine Borel Theorem, Connected Set

Lecture 14 - Tutorial II

Lecture 15 - Concept of Limit of a Sequence

Lecture 16 - Some Important Limits, Ratio Tests for Sequences of Real Numbers

Lecture 17 - Cauchy Theorems on Limit of Sequences with Examples

Lecture 18 - Fundamental Theorem on Limits, Bolzano-Weierstrass Theorem

Lecture 19 - Theorems on Convergent and Divergent Sequences

Lecture 20 - Cauchy Sequence and its Properties

Lecture 21 - Infinite Series of Real Numbers

Lecture 22 - Comparison Tests for Series, Absolutely Convergent and Conditional Convergent Series

Lecture 23 - Tests for Absolutely Convergent Series

Lecture 24 - Raabe's Test, Limit on Functions, Cluster Point

Lecture 25 - Some Results on Limit of Functions

Lecture 26 - Limit Theorems for Functions

Lecture 27 - Extension of Limit Concept (One Sided Limits)

Lecture 28 - Continuity of Functions

Lecture 29 - Properties of Continuous Functions

Lecture 30 - Boundedness Theorem, Max-Min Theorem, and Bolzano's Theorem

Lecture 31 - Uniform Continuity and Absolute Continuity

Lecture 32 - Types of Discontinuities, Continuity and Compactness

Lecture 33 - Continuity and Compactness (cont.), Connectedness

Lecture 34 - Differentiability of Real Valued Function, Mean Value Theorem

Lecture 35 - Mean Value Theorem (cont.)

Lecture 36 - Application of Mean Value Theorem, Darboux Theorem, L'Hospital's Rule

Lecture 37 - L'Hospital's Rule and Taylor's Theorem

Lecture 38 - Tutorial III

Lecture 39 - Riemann/Riemann-Stieltjes Integral

Lecture 40 - Existence of Riemann-Stieltjes Integral

Lecture 41 - Properties of Riemann-Stieltjes Integral

Lecture 42 - Properties of Riemann-Stieltjes Integral (cont.)

Lecture 43 - Definite and Indefinite Integral

Lecture 44 - Fundamental Theorems of Integral Calculus

Lecture 45 - Improper Integrals

Lecture 46 - Convergence Test for Improper Integrals