Description
Book SynopsisSuitable for undergraduates studying real analysis, this book presents the theory behind calculus directly from the basic concepts of real numbers, limits, and open and closed sets in $\mathbb{R}^n$. It gives the three characterizations of continuity: via epsilon-delta, sequences, and open sets.
Table of ContentsPart I: Real numbers and limits: Numbers and logic Infinity Sequences Functions and limits Part II: Topology: Open and closed sets Continuous functions Composition of functions Subsequences Compactness Existence of maximum Uniform continuity Connected sets and the intermediate value theorem The Cantor set and fractals Part III: Calculus: The derivative and the mean value theorem The Riemann integral The fundamental theorem of calculus Sequences of functions The Lebesgue theory Infinite series $\sum a_n$ Absolute convergence Power series Fourier series Strings and springs Convergence of Fourier series The exponential function Volumes of $n$-balls and the gamma function Part IV: Metric spaces: Metric spaces Analysis on metric spaces Compactness in metric spaces Ascoli's theorem Partial solutions to exercises Greek letters Index.
