Real Analysis

Real Analysis Curriculum

This curriculum provides a structured overview of the study of real analysis.

Foundations

The Real Number System:

• Axiomatic approach to real numbers
• Ordered field properties
• Completeness axiom (least upper bound property)
• Archimedean property
• Density of rational and irrational numbers

Sequences and Series:

• Convergence of sequences
• Limit theorems for sequences
• Monotone sequences and Cauchy sequences
• Convergence of series
• Tests for convergence (comparison, ratio, root)

Topology of the Real Line:

• Open and closed sets
• Limit points and isolated points
• Compact sets
• Connected sets
• The Heine-Borel Theorem

Core Concepts

Continuity:

• Definition of continuity
• Properties of continuous functions
• Uniform continuity
• Theorems on continuous functions (Extreme Value Theorem, Intermediate Value Theorem)
• Discontinuity

Differentiation:

• Definition of the derivative
• Theorems on differentiable functions (Mean Value Theorem, Chain Rule)
• Higher-order derivatives
• Taylor's Theorem
• L'Hôpital's Rule

Integration:

• Riemann integral
• Integrability of continuous and monotone functions
• The Fundamental Theorem of Calculus
• Improper integrals
• Applications of integration

Further Topics

Sequences and Series of Functions:

• Pointwise and uniform convergence
• The Weierstrass M-test
• Continuity, differentiability, and integrability of limit functions
• Power series
• Taylor series

Functions of Several Variables:

• Limits and continuity
• Partial derivatives
• The total derivative
• Chain rule
• Optimization