Real analysis
Undergraduate · Math
Syllabus focus
Topics typically covered
Click a topic for the full text and related unit practice.
Undergraduate Real Analysis — scope drawn from open analysis notes (e.g. Abbott-style / LibreTexts real analysis sequences) and typical US intro real analysis syllabi.
Real numbers and sequences
- Ordered fields — Ordered fields
- The axioms of R — The axioms of R
- Sequences of real numbers — Sequences of real numbers
- Convergence — Convergence
- Subsequences — Subsequences
- The Bolzano–Weierstrass theorem — The Bolzano–Weierstrass theorem
- Series of real numbers — Series of real numbers; absolute and conditional convergence
Topology and continuity
- Topology of R: open and closed sets, compactness, and the Heine–Borel theorem — Topology of R: open and closed sets, compactness, and the Heine–Borel theorem
- Limits of functions — Limits of functions; sequential characterization of limits
- Continuity — Continuity
- Intermediate value theorems (with proofs) — Intermediate value theorems (with proofs)
- Uniform continuity — Uniform continuity
- Its distinction from pointwise continuity — Its distinction from pointwise continuity
Differentiation and integration
- Differentiation — Differentiation
- Taylor's theorem — Taylor's theorem
- Riemann integration — Riemann integration
- Lower sums, integrability criteria — Lower sums, integrability criteria
- Fundamental Theorem of Calculus in the rigorous setting — Fundamental Theorem of Calculus in the rigorous setting
Advanced topics
- Sequences and series of functions — Sequences and series of functions; pointwise vs uniform convergence
- Power series and analytic functions (introduction) — Power series and analytic functions (introduction)
- Metric spaces (optional capstone): definitions and examples — Metric spaces (optional capstone): definitions and examples
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Real numbers and sequences
Ordered fields
Coming soon - Topology and continuity
Topology of R: open and closed sets, compactness, and the Heine–Borel theorem
Coming soon - Differentiation and integration
Differentiation
Coming soon - Advanced topics
Sequences and series of functions; pointwise vs uniform convergence
Coming soon
Notes
Undergraduate Real Analysis — scope drawn from open analysis notes (e.g. Abbott-style / LibreTexts real analysis sequences) and typical US intro real analysis syllabi. Topic outline: `content/topics/undergraduate/real_analysis.json`.