Mathematical Analysis I


Students should acquire the necessary knowledge and skills to pursue their learning on subsequent Analysis courses, Probability Theory, Numerical Analysis as well as to the specific disciplines of their academic studies.

General characterization





Responsible teacher

José Maria Nunes de Almeida Gonçalves Gomes


Weekly - 6

Total - 72

Teaching language



Students should have a solid background on mathematical contents taught at high-school level, within the European Union.


The text course is available CLIP. The following bibliography may also be considered:

Alves de Sá, A. e Louro, B., Cálculo Diferencial e Integral em ℝ

Alves de Sá, A. e Louro, B., Cálculo Diferencial e Integral em ℝ, Exercícios Resolvidos, Vol. 1,2,3

Anton, Bivens and Davis, Calculus ed Wiley.

Campos Ferreira, J., Introdução à Análise Matemática, ed Fundação Calouste Gulbenkian.

Lages de Lima, E., Curso de Análise Vol 1, ed IMPA (projeto Euclides)

 Rudin, Principles of Mathematical Analysis, ed Mac Graw Hill

Hairer E. Wanner G. Analysis by Its History, Springer. 

Bento de Jesus Caraça, Conceitos Fundamentais da Matemática.

Note: Tests and exams of previous editions are also available in CLIP and are usefull tools in the student preparation.

Teaching method

Teaching Method bases on conferences and problems solving sessions complemented by an individual attending schedule.

Evaluation method


In order to be evaluated, the student must attend at least 2/3 of the solving sessions.


Evaluation Methods:

1-Continuous evaluation

The continuos evaluation consists on three tests during the semester. One of the tests may be improved in the final examination date. The final grade is the average of the grades of the three tests. The student is aproved if the final grade is greater or equal than 9,5.

Each test lasts for 1h 30min. 

2-Final exam evaluation.

The student is aproved if the grade of the final exam is greater or equal than 9,5.

The final exam lasts for 3h.


Calculatory devices including mobile phones are not allowed in tests or exams.

Subject matter

1. Basic topology of the real line.

1.1 Neighborhood of a Point. Interior, exterior, boundary, isolated, limit point and limit point of a set.
1.2 Open, closed, bounded and compact sets.

2. Mathematical induction and sequences

2.1 Mathematical Induction.

2.2 Limit of a sequence. Algebra of  limits. Subsequences and sublimits. Squeezed sequence convergence theorem. Fundamental theorems. Cauchy''''s sequences.

3. Limits and Continuity in R

3.1 Definition of limit according to Cauchy and to Heine. Algebra of  limits.

3.2 Continuity of a function at a point. Continuous extension of a function. Bolzano''s theorem and Weierstrass''s theorem. Continuity of composed functions and of the inverse function (for the composition). Classical inverse functions.

4. Differential Calculus in R

4.1 Definition of the derivative of a function at a point and geometrical meaning. Derivative of a function. Derivative of the composite function and of the inverse function. Rolle''s Theorem, Lagrange''s theorem. Derivative and monotony. Darboux''s theorem and Cauchy''s theorem. L''Hospital-Cauchy rule.

4.2 Taylor''''s theorem and applications.

5. Integral Calculus in R

5.1 Primitive of a function. Primitivation by parts. Primitives by substitution. Primitives of rational functions. Primitives of irrational functions and transcendent functions.

5.2 Riemann integral. Mean value theorem for integrals. Fundamental Theorem of  Calculus. Barrow''''s rule. Integration by parts and integration by substitution. Application to the calculation of areas.

5.3 Improper integrals.