Curriculum Outline for Undergraduate Real Analysis

1. Introduction to Real Analysis

   • The Real Number System
   • Supremum and Infimum
   • Sequences and Series of Real Numbers

2. Limits and Continuity

   • Limits of Sequences
   • Limits of Functions
   • Continuous Functions

3. Differentiation

   • The Derivative
   • Differentiation Rules
   • The Mean Value Theorem and Applications
   • L'Hôpital's Rule

4. Integration

   • The Riemann Integral
   • Properties of the Integral
   • Fundamental Theorem of Calculus
   • Improper Integrals

5. Sequences and Series of Functions

   • Pointwise and Uniform Convergence
   • Power Series and Radius of Convergence
   • Functions Defined by Series

6. Metric Spaces

   • Open, Closed, and Compact Sets
   • Convergence in Metric Spaces
   • Continuity in Metric Spaces

7. Topology of the Real Numbers

   • Open and Closed Sets in (\mathbb{R})
   • Compactness and Connectedness
   • Completeness

8. Advanced Topics (Optional)

   • Differentiation and Integration in (\mathbb{R}^n)
   • Lebesgue Measure and Integral
   • Fourier Analysis