Course description: The most important concepts, methods and applications of real analysis and the theory of metric spaces are covered, with an emphasis on examples and problem solving. The course is self-contained, the only prerequisite is calculus. We do not follow closely a single textbook, but for those who wish to consult, the relevant chapter numbers of the reference books will be given.

Topics:

• Review: Real numbers, numerical sequences.
• Differentiation I.: Limit, continuity and differentiation of single variable, real functions. Mean value theorems. Applications.
• Integration: Indefinite and Riemann integrals, inequalities, estimating sums with integrals.
• Metric spaces: Euclidean spaces. Topology, convergence, continuity, compactness, connectedness, completeness, separability. The metric space C([a,b]).
• Differentiation II.: Derivation of functions of several variables.
• Infinite series: Numerical series, sequences and series of functions, power series and analytic functions.