Course 2024-2025 a.y.

• Real and complex numbers.

• Sequences: convergence, subsequences, limsup and liminf. Cardinality: countable and uncountable sets. Discrete processes.
• Series: convergence and absolute convergence, elementary and advanced tests, infinite sums, operations on series.
• Limit and continuity of real functions with their main applications. Uniform continuity.
• Differential calculus for real functions of one variable: derivatives, mean value theorems and their applications, l’Hopital’s rule, Taylor expansions, convexity.
• Cauchy-Riemann integral calculus: properties of the integral, integrability of monotonic and continuous functions, the Fundamental Theorem of Calculus, integration by parts, change of variables.
• Improper integrals, integrals and series, integral function.

• Demonstrate a comprehensive understanding of the language and of the fundamental notions and results of the mathematical theory of one-variable real functions, including sequences, limits, series, continuity, differential and integral calculus.
• Express these notions in a conceptually and formally correct way, using adequate definitions, theorems, and proofs.
• Demonstrate familiarity with the analytic tools and methods needed to solve simple problems and to investigate the behaviour of real functions and discrete processes.

• Apply the fundamental results and techniques of mathematical analysis to the solution of problems and exercises.
• Actively search for links between the properties of mathematical objects and to solve assigned problems.
• Formulate simple problems through mathematical models, which can be studied with the help of calculus and analytical tools.
• Adapt already studied arguments to identify and demonstrate simple variants of familiar mathematical statements.

• Lectures
• Practical Exercises
• Individual works / Assignments

Online lectures have the same conceptual role as face-to-face lectures. The actual blend of face-to-face lectures and online lectures will mainly depend on external constraints.

Exercise sessions (again: both face-to face and online) are dedicated to the application of the main theoretical results obtained during lectures to problems and exercises of various nature.

Students will presumably be evaluated on the basis of written and oral exams, which can be taken in one of the two following ways.

• The exam can be split in two partial exams (October and January). Each partial may contain multiple-choice, numeric and open-answer questions; the second partial may also involve an oral exam; each partial weighs for at least one-third of the final mark. There could also be online tests/assignments, the total of which weighs no more than one sixth of the final grade. Each type of questions contributes in a specific way to the assessment of the students' acquired knowledge. Multiple-choice and numeric questions mainly aim at evaluating the knowledge of the fundamental mathematical notions and the ability to apply these notions to the solution of simple problems and exercises, while open-answer questions and oral exams mainly aim at evaluating:
  • The ability to articulate the knowledge of mathematical notions in a conceptually and formally correct way, adequately using definitions, theorems and proofs.
  • The ability to actively search for possible links between the properties of mathematical objects.
  • The ability to apply mathematical notions to the solution of more complex problems and exercises.  
• The exam can also be taken as a single general exam, which contains both multiple-choice questions and open-answer questions, and may also involve an oral exam. The general exam covers the whole syllabus of the course and it can be taken in one of the four general sessions scheduled in the academic year (the two regular sessions in January and January/February, or the two make-up sessions in June/July and August/September). This option is mainly meant for students who have withdrawn from the two-partials procedure or could not follow it. Each type of questions contributes in a specific way to the assessment of the students' acquired knowledge. Multiple-choice and numeric questions mainly aim at evaluating the knowledge of the fundamental mathematical notions and the ability to apply these notions to the solution of simple problems and exercises, while open-answer questions and oral exams mainly aim at evaluating:
  • The ability to articulate the knowledge of mathematical notions in a conceptually and formally correct way, adequately using definitions, theorems and proofs. 
  • The ability to actively search for possible links between the properties of mathematical objects.
  • The ability to apply mathematical notions to the solution of more complex problems and exercises.

• Claudio Canuto, Anita Tabacco: Mathematical Analysis 1
  Pearson  2002. ISBN 9788891931115 (printed edition) ISBN 9788891931122 (digital edition).

• Mauro D’Amico, Jacopo De Tullio, Guido Osimo, Giacomo Enrico Sodini: Mathematical Analysis - Module 1 Exercises
  BAI Series, volume 1, Università Bocconi. EGEA, Academic Year 2021/2022. ISBN: 9788864074498 (Printed edition) 9788864074504 (eBook)

• Integrative teaching materials (slide, notes, exercises)

• (to learn more about the course content) Laczkovich, Miklós, T. Sós, Vera: Real Analysis. Foundations and Functions of One Variable

  Series: Undergraduate Texts in Mathematics. Springer Verlag, New York, 2015. x+483 pp. ISBN: 978-1-4939-4222-0 (Softcover edition) 978-1-4939-2766-1