At the end of the course:
- the students will acquire the basic notions in Mathematical Analysis. In
particular, the
theory of limits and the differential calculus for functions of one real
variAt the end of the course the students will know the fundamental
results of differential and integral calculus.
- the students will be able to solve simple exercises on this topic and also
they will be able to produce elementary proofs of some simple properties
of the real functions depending on one
space variable.
- the students will be able to handle the usual reasoning
tools in calculus (limit calculus, derivation, integration)
- the students will be able to express themselves in a correct way on
elementary topics from Mathematical Analysis.
- the students will be able to use the handbooks in this discipline.

Elementary results in logic (deductive reasoning). Elementary results in
algebra and analytic geomety (high school level). Notion on functions and
relations.

1. Natural numbers. Induction principle. The binomial theorem. Integers and rational numbers. 2. Real numbers. The axiom of separation. Existence of the supremum. Density of rationals in reals. Cantor's theorem on boxed intervals. The Bolzano-Weierstrass theorem. Complex numbers. 3. Metric spaces. Neighborhoods, open and closed sets. Cluster points. Boundary of a set. 4. Continuous functions between metric spaces. Bolzano's theorem. 5. Limits of functions between metric spaces. 6. Weierstrass' theorem. Compacts of R and their characterization. Cauchy sequences. Completeness of R. Heine's theorem. 7. Differential calculus for real functions of one real variable. Higher order derivatives. Rules of derivation: sum, product, quotient, composite functions, inverses. Rolle, Lagrange and Cauchy theorems. Rules of de l'Hopital. Characterization of monotone functions. Convex and concave functions. Taylor's formula with Lagrange remainder. 8. Integral calculus for real functions of one real variable. Integrability of continuous functions. Fundamental theorem of differential and integral calculus. Integration by parts and by substitution.

- G. Prodi, Analisi Matematica, Ed. Boringhieri.
- C.D. Pagani, S. Salsa, Analisi Matematica 1, Ed. Zanichelli.
- E. Giusti, Analisi Matematica 1, Ed. Boringhieri.
- E. Giusti, Esercizi e Complementi di Analisi Matematica, vol.1, Ed. Boringhieri.

1. Natural numbers. Induction principle. The binomial theorem. Integers and rational numbers.

2. Real numbers. The axiom of separation. Existence of the supremum. Density of rationals in reals. Cantor's theorem on boxed intervals. The Bolzano-Weierstrass theorem. Complex numbers.

3. Metric spaces. Neighborhoods, open and closed sets. Cluster points. Boundary of a set.

4. Continuous functions between metric spaces. Bolzano's theorem.

5. Limits of functions between metric spaces.

6. Weierstrass' theorem. Compacts of R and their characterization. Cauchy sequences. Completeness of R. Heine's theorem.

7. Differential calculus for real functions of one real variable. Higher order derivatives. Rules of derivation: sum, product, quotient, composite functions, inverses. Rolle, Lagrange and Cauchy theorems. Rules of de l'Hopital. Characterization of monotone functions. Convex and concave functions. Taylor's formula with Lagrange remainder.

8. Integral calculus for real functions of one real variable. Integrability of continuous functions. Fundamental theorem of differential and integral calculus. Integration by parts and by substitution.

Lectures. Classworks and homeworks. A tutor (phd student) will help the students in homeworks and will coordinate study groups.

Any possible changes to what described here will be communicated on Moodle.

Final Exam. Students scoring 25 or higher are eligible for an optional oral exam, which can either improve or lower their grade. Students who choose to confirm their written exam score can do so for a maximum grade of 27/30