Mathematical Analysis I


Domain of the basic techniques required for the Mathematical Analysis of real functions of real variable.

The students should acquire not only calculus skills, mandatory to the acquisition of some of the knowledge lectured in Physics, Chemistry and other Engineering subjects, but also to develop methods of solid logic reasoning and analysis.

Being a first course in Mathematical Analysis, it introduces some of the concepts which will be deeply analyzed and generalized in subsequent courses.

General characterization





Responsible teacher

Ana Luísa da Graça Batista Custódio


Weekly - 5

Total - 60

Teaching language



The student must master the mathematical knowledge lectured until the end of Portuguese high school teaching.


Recommended Bibliography

  1. Ana Alves de Sá e Bento Louro, Cálculo Diferencial e Integral em R
  2. Jaime Campos Ferreira, Introdução à Análise Matemática, Fundação Calouste Gulbenkian, 1982
  3. Carlos Sarrico, Análise Matemática, Leituras e Exercícios, Gradiva, 1997
  4. Robert G. Bartle and Donald R. Sherbert, Introduction to Real Analysis, John Wiley & Sons Inc., 1999
  5. Rod Haggarty, Fundamentals of Mathematical Analysis, Prentice Hall, 1993


Teaching method

Theoretical classes consist in a theoretical exposition of subjects, illustrated by application examples.

Practical classes consist in the resolution of some application exercises for the methods and results lectured in the theoretical classes, as well as support for exercises solved by the students in autonomous work.

Any questions or doubts will be addressed during the classes, during the weekly sessions specially programmed to attend students or in individual sessions previously scheduled between professors and students.

Evaluation method

Please contact Professor Ana Luísa Custódio (

Subject matter

1. Topology - Mathematical Induction - Sequences

Basic topology of the real numbers. Order relation.

Mathematical induction.

Generalities about sequences. Convergence of a sequence and properties for calculus of limits. Subsequences. Bolzano-Weierstrass theorem. 

2. Limits and Continuity

Generalities about real functions of real variable. Convergence according to Cauchy and Heine. Calculus properties.

Continuity of a function at a given point. Properties of continuous functions. Bolzano theorem. Continuity and reciprocal bijections. Weierstrass theorem. 

3. Differenciability

Generalities. Fundamental theorems: Rolle, Lagrange and Cauchy. Calculus techniques for limits. Taylor theorem and applications.

4. Indefinite Integration

Introduction. Indefinite integration by parts. Indefinite integration by substitution.  Indefinite integration of rational functions.

5. Riemann Integration

Introduction. Fundamental theorems. Definite integration by parts and by substitution. Some applications.

Improper integration.