1.The Spaces R, Rk, and C
1.1 The Real Numbers R
1.2 The Real Spaces Rk
1.3 he Complex Numbers C
2. Point-Set Topology
2.1 Bounded Sets
2.2 Classification of Points
2.3 Open and Closed Sets
2.4 Nested Intervals and the Bolzano-Weierstrass Theorem
2.5 Compactness and Connectedness
3. Limits and Convergence
3.1 Definitions and First Properties
3.2 Convergence Results for Sequences
3.3 Topological Results for Sequences
3.4 Properties of Infinite Series
3.5 Manipulations of Series in R
4. Functions: Definitions and Limits
4.1 Definitions
4.2 Functions as Mappings
4.3 Some Elementary Complex Functions
4.4 Limits of Functions
5. Functions: Continuity and Convergence
5.1 Continuity
5.2 Uniform Continuity
5.3 Sequences and Series of Functions
6. The Derivative
6.1 The Derivative for f : D1 → R
6.2 The Derivative for f : Dk → R
6.3 The Derivative for f : Dk → Rp
6.4 The Derivative for f : D → C
6.5 The Inverse and Implicit Function Theorems
Hints and Solutions to Odd Embedded Exercises
Biography
Christopher Apelian completed a Ph.D. in mathematics in 1993 at New YorkUniversity’s Courant Institute of Mathematical Sciences and then joined the Department of Mathematics and Computer Science at Drew University. He has published papers in the applications of probability and stochastic processes to the modeling of turbulent transport.
Steve Surace joined Drew University’s Department of Mathematics and Computer Science in 1987 after earning his Ph.D. in mathematics from New York University’s Courant Institute. His mathematical interests include analysis, mathematical physics, and cosmology.
