Università Roma Tre

Part 1: Axiomatics of R and its main subsets Axiomatic definition of R.
Inductive assemblies; definition of N and induction principle. Definition of Z
and Q; Z is a ring, Q is a field.
Nth roots; rational powers.

Part 2: Theory of Limits
The extended line R*: intervals, neighbourhoods and accumulation points.
Limits of functions in R*.
Comparison theorems.
Lateral limits; limits of monotone functions.
Algebra of limits on R and R*.
Composition limit of functions.
Limits of inverse functions.
Notable limits. The number of Napier. Exponential and trigonometric functions.

Part 3: Continuous functions
Topology of R.
Theorem of existence of zeroes. Bolzano-Weierstrass theorems. Weierstrass’ Theorem.
Uniformly continuous functions.

Part 4: Differentiable functions
Rules of derivation. Derivatives of elementary functions.
Local minima and maxima and elementary theorems on derivatives (Fermat, Rolle, Cauchy,
Lagrange).
Bernoulli-Hopital theorem.
Convexity.
Taylor’s formulae.

Part 5: Riemann integral in R.
The Riemann integral and its fundamental properties.
Integration criteria. Integrability of continuous and monotone functions.
The fundamental theorem of calculus and its applications (integration by parts,
changes of variables in integration). Generalized ("improper") integrals and
related integrability criteria.

Luigi Chierchia: Corso di analisi. Prima parte. Una introduzione rigorosa all'analisi matematica su R
McGraw-Hill Education Collana: Collana di istruzione scientifica
Data di Pubblicazione: giugno 2019
EAN: 9788838695438 ISBN: 8838695431
Pagine: XI-374 Formato: brossura
https://www.mheducation.it/9788838695438-italy-corso-di-analisi-prima-parte

Testi di esercizi:
Giusti, E.: Esercizi e complementi di Analisi Matematica, Volume Primo, Bollati Boringhieri, 2000
Demidovich, B.P., Esercizi e problemi di Analisi Matematica, Editori Riuniti, 2010

Open, closed, compact sets. Weierstrass theorem. Uniformly continuous functions.
Differentiability, the derivative and its interpretations. Rules for computing derivatives. Derivatives and monotonicity. Fundamental theorems on differentiability (Fermat, Rolle, Cauchy, Lagrange). Bernoulli-Hopital theorems. Critical points. Second derivative. Convex functions. Qualitative study of functions. Higher order derivatives and Taylor's formula (Peano's theorem). Use of Taylor's formula in limits.
The Riemann integral: partial sums, integrability. Classes of integrable functions (monotone functions, continuous and piecewise functions). Computation of primitives. The fundamental theorem of calculus. Integral remainder in Taylor's formula. Improper integrals; comparison with series.
Complex numbers, exponential series in the complex plane and fundamental theorem of algebra.

Luigi Chierchia, Corso di Analisi, prima parte, Una introduzione rigorosa all'analisi matematica su R.