Real Analysis I


To know and understand fundamental concepts of Analysis in one dimension.

General characterization





Responsible teacher

Ana Margarida Fernandes Ribeiro


Weekly - 4

Total - 57

Teaching language



Available soon


J. Campos Ferreira, Introdução à Análise Matemática, Fundação Calouste Gulbenkian, 1982.

M. Figueira, Fundamentos de Análise Infinitesimal, Textos de matemática, Dep. de matemática da FCUL, 1996.

K. Ross, Elementary Analysis The Theory of Calculus, Second Edition, Springer, 2013.

E. Lages Lima, Curso de Análise vol. 1, 14ª ed., IMPA, 2012. M. Spivak, Calculus, 3rd edition, Publish or Perish Inc., 2006.

Teaching method

Classes are theoretical/practical with oral presentation of concepts and results together with their proofs and complemented with examples and applications. A list of exercises that includes several theoretical problems is provided to the students to be solved independently. Specific student difficulties will be addressed during classes or in individual sessions scheduled with the professor. Continuous evaluation is based on two tests. If a student does not obtain approval through continuous evaluation, he can try it in an additional assessment.

Evaluation method

The evaluation is a mark between 0 and 20 values and the student passes if the final mark is above or equal to 9,5.

There are two evaluation items:

1) a mid-term test (corresponding to 50% of the final mark);
2) a second test at the end of the classes period (corresponding to 50% of the final mark).

If the student fails in the continuous evaluation, the student must pass an exam. More detailed rules are available in the portuguese version.

Subject matter

1. Supremum axiom and density of rational and irrational numbers on R. Topological notions. 2. Sequences of real numbers: limit definition, review on convergence results. Subsequences, superior and inferior limit of a sequence, Bolzano Weierstrass theorem. Cauchy sequences. 3. Series of real numbers: series with nonnegative terms, series with alternating signs, absolute convergence. 4. Functions of real variable: Cauchy and Heine limit definitions, inverse function theorem, Weierstrass theorem, fixed point Picard theorem, uniform continuity, Cantor theorem. 5. Sequences of functions: pointwise and uniform convergence, limits of sequences of continuous functions, limits of sequences of integrals (observing the rigorous construction of Riemann integral). 6. Series of functions: pointwise and uniform convergence, power series, analytic functions and Taylor series.