| Topic Number | Major Topic | Text Section | Subtopics |
| 1 |
Completeness |
2.6 | The Cauchy Criterion and Completeness |
| 2 |
The Basic Topology of R |
3.1 | The Cantor Set |
| 3.2 | Open and Closed Sets |
| 3.3 | Compact Sets |
| 3.4 | Perfect Sets and Connected Sets |
| 3.5 | Baire's Theorem |
| 3 |
Functional Limits and Continuity |
4.1 | The Examples of Dirichlet and Thomae |
| 4.2 | Functional Limits |
| 4.3 | Combinations of Continuous Functions |
| 4.4 | Continuous Functions on Compact Sets |
| 4.5 | The Intermediate Value Theorem |
| 4.6 | Sets of Discontinuity |
| 4 |
The Derivative |
5.1 | Are Derivatives Continuous? |
| 5.2 | Derivatives and the Intermediate Value Theorem |
| 5.3 | The Mean Value Theorem |
| 5.4 | A Continuous Nowhere-Differentiable Function |
| 5 |
Sequences and Series of Functions |
6.1 | Branching Processes |
| 6.2 | Derivatives and the Intermediate Value Theorem |
| 6.3 | Uniform Convergence and Differentiation |
| 6.4 | Series of Functions |
| 6.5 | Power Series |
| 6.6 | Taylor Series |
| 6 |
The Riemann Integral |
7.1 | How Should Integration be Defined? |
| 7.2 | The Definition of the Riemann Integral |
| 7.3 | Integrating Functions with Discontinuities |
| 7.4 | Properties of the Integral |
| 7.5 | The Fundamental Theorem of Calculus |
| 7.6 | Lebesgue's Criterion for Riemann Integrability |
| 7 |
Additional Topics |
8.1 | The Generalized Riemann Integral |
| 8.2 | Metric Spaces and the Baire Category Theorem |
| 8.3 | Fourier Series |
| 8.4 | A Construction of R from Q |