Contents
1 Topology of the Real Line
1.1 Intervals
1.2 Closed Sets
1.3 Open Sets
1.4 Compact Sets
1.5 Cantor’s Intersection or Nested Interval theorem
1.6 Connected Sets
1.7 Practice Problems
2 Continuous functions
2.1 Limits of functions
2.2 Continuous functions
2.3 Properties of Continuous functions
2.4 Continuity and Compactness
2.5 Continuity and Connectedness
2.6 Uniform Continuity
2.7 Uniform Convergence
2.8 Sequence of functions
2.9 Approximation
2.10 Monotone and Inverse functions
2.11 Practice Problems
3 Functions of Bounded Variation
3.1 Algebraic Properties
4 Differentiable functions
4.1 Differentiability and Continuity
4.2 Rolle’s Theorem And The Mean Value Theorem (MVT)
4.3 Local Maximum and Local Minimum
4.4 Higher Order Derivatives
4.5 Taylor’s Theorem
4.6 L’H\(\hat {\text {o}}\)pital’s Rule
4.7 Practice Problems
5 The Riemann Integral
5.1 Partitions and the Riemann Sums
5.2 Properties of the Riemann Integral
5.3 Fundamental Theorem of Calculus
5.4 Improper Integrals
5.5 Riemann-Stieljes Integral
References
5.6 Practice Problems
1.1 Intervals
1.2 Closed Sets
1.3 Open Sets
1.4 Compact Sets
1.5 Cantor’s Intersection or Nested Interval theorem
1.6 Connected Sets
1.7 Practice Problems
2 Continuous functions
2.1 Limits of functions
2.2 Continuous functions
2.3 Properties of Continuous functions
2.4 Continuity and Compactness
2.5 Continuity and Connectedness
2.6 Uniform Continuity
2.7 Uniform Convergence
2.8 Sequence of functions
2.9 Approximation
2.10 Monotone and Inverse functions
2.11 Practice Problems
3 Functions of Bounded Variation
3.1 Algebraic Properties
4 Differentiable functions
4.1 Differentiability and Continuity
4.2 Rolle’s Theorem And The Mean Value Theorem (MVT)
4.3 Local Maximum and Local Minimum
4.4 Higher Order Derivatives
4.5 Taylor’s Theorem
4.6 L’H\(\hat {\text {o}}\)pital’s Rule
4.7 Practice Problems
5 The Riemann Integral
5.1 Partitions and the Riemann Sums
5.2 Properties of the Riemann Integral
5.3 Fundamental Theorem of Calculus
5.4 Improper Integrals
5.5 Riemann-Stieljes Integral
References
5.6 Practice Problems