Learn · A Basic Course in Real Analysis
Lecture 31 of 46
Uniform Continuity and Absolute Continuity
Prof. P.D. Srivastava · IIT Kharagpur
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- A Basic Course in Real Analysis, IIT Kharagpur (at NPTEL)
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only.
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All the lectures
- Rational Numbers and Rational Cuts
- Irrational numbers, Dedekind's Theorem
- Continuum and Exercises
- Continuum and Exercises (Contd.)
- Cantor's Theory of Irrational Numbers
- Cantor's Theory of Irrational Numbers (Contd.)
- Equivalence of Dedekind and Cantor's Theory
- Finite, Infinite, Countable and Uncountable Sets of Real Numbers
- Types of Sets with Examples, Metric Space
- Various properties of open set, closure of a set
- Ordered set, Least upper bound, greatest lower bound of a set
- Compact Sets and its properties
- Weiersstrass Theorem, Heine Borel Theorem, Connected set
- Tutorial - II
- Concept of limit of a sequence
- Some Important limits, Ratio tests for sequences of Real Numbers
- Cauchy theorems on limit of sequences with examples
- Fundamental theorems on limits, Bolzano-Weiersstrass Theorem
- Theorems on Convergent and divergent sequences
- Cauchy sequence and its properties
- Infinite series of real numbers
- Comparison tests for series, Absolutely convergent and Conditional convergent series
- Tests for absolutely convergent series
- Raabe's test, limit of functions, Cluster point
- Some results on limit of functions
- Limit Theorems for functions
- Extension of limit concept (one sided limits)
- Continuity of Functions
- Properties of Continuous Functions
- Boundedness Theorem, Max-Min Theorem and Bolzano's theorem
- Uniform Continuity and Absolute Continuity
- Types of Discontinuities, Continuity and Compactness
- Continuity and Compactness (Contd.), Connectedness
- Differentiability of real valued function, Mean Value Theorem
- Mean Value Theorem (Contd.)
- Application of MVT , Darboux Theorem, L Hospital Rule
- L'Hospital Rule and Taylor's Theorem
- Tutorial - III
- Riemann/Riemann Stieltjes Integral
- Existence of Reimann Stieltjes Integral
- Properties of Reimann Stieltjes Integral
- Properties of Reimann Stieltjes Integral (Contd.)
- Definite and Indefinite Integral
- Fundamental Theorems of Integral Calculus
- Improper Integrals
- Convergence Test for Improper Integrals