Hexa Scholar
MathAnalysis and pure mathematics

Real analysis

Limits, convergence, continuity and integration, stated precisely.

1 lesson open · 13 planned

Pages you can open

The ε–δ definition of a limit What does it mean for f(x) to get close to L as x gets close to a? ~25 min

Planned

In learning order. These are not pages yet.

  1. Convergent sequences and the ε–N gameIntro
  2. Completeness: why the real line has no holesIntro
  3. Riemann vs Lebesgue: slicing the area two waysAdvanced undergrad
  4. Cauchy sequences: converging without knowing the limitAdvanced undergrad
  5. Series convergence tests, seen as areasIntro
  6. Continuity and the intermediate value theoremIntro
  7. Uniform continuity: one δ for the whole domainAdvanced undergrad
  8. Pointwise vs uniform convergence of functionsAdvanced undergrad
  9. Measure zero and the Cantor setAdvanced undergrad
  10. Compactness: why closed and bounded mattersAdvanced undergrad
  11. The contraction mapping theoremAdvanced undergrad
  12. The mean value theorem: somewhere the slope equals the averageIntro
  13. When can you swap a limit and an integralAdvanced undergrad

How it connects

Builds on Proof, logic and sets Calculus

Leads to Probability Topology Differential equations Numerical methods Corporate finance Stochastic processes connection

Subjects marked “connection” are linked by a related idea, not by a prerequisite.

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