§ Real Analysis

Topics

The real numbers

• The real numbers
  • Order relations on sets and fields
  • Completeness
  • Decimal representation of reals

Theorems:

• (Theorem) There exists a unique isomorphism between two complete ordered fields
• (Theorem) Between any two real numbers is a rational number

Basic topology of metric spaces

• Metrics, metric spaces, and the metric topology
• Open sets
• Closed sets and closures
• Bounded sets and functions
• Compactness
• (Path-)connectedness

Theorems:

• (Theorem) Heine-Borel
• (Theorem) Compactness, limit point compactness, and sequential compactness are equivalent on metric spaces

Sequences and series

• Sequences
• Cauchy sequences and complete metric spaces
• Limits and accumulation points
• Series

Continuity

• Pointwise continuity
• Continuous functions

Theorems:

• (Theorem) Intermediate value
• (Theorem) Continuous functions on compact sets are uniformly continuous

Differentiability

• Derivatives of real functions
• Local extrema of real functions

Theorems:

• (Theorem) Rolle’s and mean value
• (Theorem) L’Hopital’s rule

Integration

Lebesgue measure and integral

• Lebesgue measure
• Measurable sets
• Measurable functions
• Lebesgue integral

Theorems:

• (Theorem) Approximation of measurable sets

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Notes

• Continuity – continuity, limits, uniform continuity, monotone functions
• Differentiation – derivative, mean value theorem, intermediate value property
• Sequences – sequences of functions and integration/differentiation
• Riemann integration – partitions, fundamental theorem of calculus
• Series – convergence, power series