Real Number System
Algebra of the Real Number System
Upper and Lower Bounds
LUB Property and Its Applications
Absolute Value and Triangle Inequality
Sequences and Their Convergence
Sequences and Their Convergence
Cauchy Sequences
Monotone Sequences
Sandwich Lemma
Some Important Limits
Sequences Diverging to
Subsequences
Sequences Defined Recursively
Continuity
Continuous Functions
Definition of Continuity
Intermediate Value Theorem .
Extreme Value Theorem
Monotone Functions
Limits
Uniform Continuity
Continuous Extensions
Differentiation
Differentiability of Functions
Mean Value Theorems
L'Hospital's Rules
Higher-order Derivatives
Taylor's Theorem
Convex Functions
Cauchy's Form of the Remainder
Infinite Series
Convergence of an Infinite Series
Abel's Summation by Parts
Rearrangements of an Infinite Series
Cauchy Product of Two Infinite Series
Riemann Integration
Darboux Integrability
Properties of the Integral
Fundamental Theorems of Calculus
Mean Value Theorems for Integrals
Integral Form of the Remainder
Riemann's Original Definition
Sum of an Infinite Series as an Integral
Logarithmic and Exponential Functions
Improper Riemann Integrals
Sequences and Series of Functions
Pointwise Convergence
Uniform Convergence
Consequences of Uniform Convergence
Series of Functions
Power Series
Taylor Series of a Smooth Function
Binomial Series
Weierstrass Approximation Theorem
A Quantifiers
B Limit Inferior and Limit Superior
C Topics for Student Seminars
D Hints for Selected Exercises
Bibliography
Index
Biography
Dr. Ajit Kumar is a faculty member at the Institute of Chemical Technology, Mumbai, India. His main interests are differential geometry, optimization and the use of technology in teaching mathematics. He received his Ph.D. from University of Mumbai. He has initiated a lot of mathematicians into the use of open source mathematics software. Dr. S Kumaresan is currently a professor at University of Hyderabad. His initial training was at Tata Institute of Fundamental Research, Mumbai where he earned his Ph.D. He then served as a professor at University of Mumbai. His main interests are harmonic analysis, differential geometry, analytical problems in geometry, and pedagogy. He has authored five books, ranging from undergraduate level to graduate level.
"… there are some unique features that put this book aside. … a welcome addition to the library of teachers and student alike."
—Zentralblatt MATH 1308"… this book describes the basic results of analysis in an extremely clear, straightforward, and well-motivated way. … if you’re looking for a text on the easy end of the spectrum for a course in real analysis, then this book is certainly worth a serious look …"
—MAA Reviews, October 2014
