1. Real Numbers
a. Numbers; Natural (ℕ), Integers (ℤ), Rational (ℚ), Complex (ℂ), Real (ℝ).
b. Basic Properties Fields;Ordered Field; Absolute Value Function; The Archimedean Property
c. Basic Set Theory and Set of Real Numbers
d. Absolute Value and Bounded Functions
2. The Extended Real Line and Its topology
a. The Extended Real Line
b. Neighbourhoods
c. Types of Points for a Set
d. Interior, Exterior, and Boundary of a Set
e. Open Sets and Closed Sets in ℝ and ℝ *
f. Closure of a Set
g. Compactness
3. Sequences and Limits
a. Sequences
b. Limit Points on a Sequence
c. Limit Points: Suprema and infima
d. Limit of a Sequence
e. Sequences and
f. Monotone Sequences and Uniqueness of the limit of a sequence
4. Series
5. Continuous Functions
a. Limits of Functions
b. Continuous Functions
c. Min-max and intermediate value theorems
d. Uniform continuity
e. Limits at Infinity
f. Monotone Functions and Continuity
6. Real Functions Derivatives
a. The Derivative
b. The Mean Value Theorem
c. Taylor‟s Theorem
d. Inverse Function Theorem
References
1. John K. Hunter, An Introduction to real analysis, 2014
2. Jiri Lebl; Introduction to Real Analysis, Volume I Volume 5.4 2021
- Teacher: JOACHIM