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Functions of a Real Variable I

Code: 107840 ECTS Credits: 6
2025/2026
Degree Type Year
Mathematics FB 1

Contact

Name:
Laura Prat Baiget
Email:
laura.prat@uab.cat

Teachers

Juan Eugenio Mateu Bennassar

Teaching groups languages

You can view this information at the end of this document.


Prerequisites

In order for a student to be able to follow the subject normally, it is essential that they have a certain skill in the algebraic manipulation of fractions, expressions containing roots and powers, resolution of linear systems and basic arithmetic of numbers and polynomials. It is also highly advisable that the student has knowledge of trigonometry. Finally, it is expected that the student can, without much difficulty, graphically represent relatively simple functions of one variable, interpret the derivative of a function and calculate relatively simple primitives. We also assume that the person taking this subject is familiar with logical reasoning and knows how to negate sentences or propositions

The most important requirement, however, is a great curiosity to understand and delve deeper into the concepts that will be studied.

Objectives and Contextualisation

So that a student can follow the subject normally, it is essential that he has some skill in the algebraic manipulation of fractions, expressions with roots and powers, solving linear systems and basic arithmetic of numbers and polynomials. It is also highly recommended that the student has knowledge of trigonometry. Finally, it is expected that the student can, without major difficulty, graphically represent relatively simple functions of a variable, interpret the derivative of a function and calculate relatively simple primitives. We also assume that the person studying this subject is familiar with logical reasoning and knows how to negate sentences or propositions.

The most important requirement, however, is a great curiosity to understand and deepen the concepts that will be studied.

Learning Outcomes

  1. CM01 (Competence) Write elementary proofs in the field of algebra and analysis in an orderly and precise manner.
  2. CM02 (Competence) Develop autonomous strategies for solving basic mathematical problems.
  3. KM01 (Knowledge) Identify the basics of linear algebra and single-variable analysis.
  4. KM02 (Knowledge) Identify the rules of differentiation and integration of functions, as well as the basic results of the differential calculus in one real variable.
  5. KM03 (Knowledge) Relate the visual properties of a function graph in a real variable to its analytical properties.
  6. SM01 (Skill) Apply the rules of algebra and single-variable analysis to the classification of applications according to various criteria (rank, determinant, Jordan forms, existence of maxima and minima, asymptotes).
  7. SM02 (Skill) Apply the basics of linear algebra and analysis to a variable to solve mathematical problems.
  8. SM03 (Skill) Relate the concepts of linear algebra to those of single-variable analysis (linearity of differential and integral operators or continuity of matrix operations, etc.).

Content

The course program is organized into three chapters:

I. The real line.

Rational numbers and their incompleteness.
Supreme and least of a set.
The concept of real number. Axiomatics. Decimal expression.
Operations and inequalities between real numbers.
Distinguished real numbers: Π and e
II. Sequences of real numbers.

Real functions of discrete or continuous variable
Limit of a sequence. Algebraic properties.
Monotone sequences.
Accumulation points. Partial sequences.
The Bolzano-Weierstrass Theorem.
Cauchy sequences and restatement of the completeness axiom.
Calculus of limits.
III. Continuity of functions.

Functions of real variable. Domain of a function.
Polynomial, rational, exponential and trigonometric functions vs. experimental functions.
Limit of a function at a point, lateral limits. Basic properties of limits. Asymptotes.
Continuity of a function.
Bolzano's theorem, location of roots.
Intermediate value theorem and Weierstrass's theorem.
Monotone functions. Inverse functions.

Activities and Methodology

Title Hours ECTS Learning Outcomes
Type: Directed      
exams preparation 11 0.44 CM01, CM02, KM01, KM02, KM03, SM01, SM02, SM03, CM01
problem solving 50 2 CM01, CM02, KM01, KM02, KM03, SM01, SM02, SM03, CM01
Theory classes 29 1.16 CM01, CM02, KM01, KM02, KM03, SM01, SM02, SM03, CM01
Type: Supervised      
problem classes 15 0.6 CM01, CM02, KM01, KM02, KM03, SM01, SM02, SM03, CM01
tutorized activities 11 0.44 CM01, CM02, KM01, KM02, KM03, SM01, SM02, SM03, CM01
Type: Autonomous      
theory study 25 1 CM01, CM02, KM01, KM02, KM03, SM01, SM02, SM03, CM01

The subject has two theory groups, two problem groups and four seminar-practice groups.

There will be two weekly sessions of one hour of theory and one problem session.

The seminars will be for tutored group work.

The timetables and classrooms must be consulted on the degree website.

In the subject's Moodle, the student will have the necessary material at their disposal to follow all the sessions. There, they will be able to find notes, problem lists, observations made by the teaching staff or news that may be relevant to the development of the subject and, eventually, other materials that are of interest to the students.

Theory classes. The teacher will develop the topics of the program in the order indicated. It is essential that the student works regularly, using the textbooks indicated in the bibliography or class notes. Sometimes the teacher will leave it up to the student to complete the demonstrations of some results, work that will have to be done individually with the help of textbooks and using the tutoring hours.

Problem classes. Three lists of problems will be distributed each semester which, as mentioned before, will be available on Moodle. In the problem class, teachers will solve or give instructions on some of the problems on the lists on the board, but not all of them, which is why it is essential that the student regularly carries out individual work based on the lists. Thinking about the problems, even if they do not arise, and dedicating time to them is essential to be able to face this subject with guarantees.

Seminars. Seminars are group work activities under the supervision of a tutor teacher. Each session will follow a script that will also be distributed in class. Of the eight seminars of the course, four of them (two in the first semester and two in the second) will incorporate an assessable activity. The dates of the assessable seminars will be announced on Moodle sufficiently in advance. In order to maintain balance between groups, it is important that students go to the problem and seminar group that has been assigned to them. Changes will only be allowed in exceptional or justified situations.

Finally, it is remembered that students will have a few hours of tutoring in the office of the theory, problem and seminar teachers, where they can ask questions and request all kinds of help with their work. The schedule for each teacher will be announced on Moodle.

Note: 15 minutes of a class will be reserved, within the calendar established by the center/degree, for students to complete the surveys to evaluate the performance of the teaching staff and the evaluation of the subject/module.

Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.


Assessment

Continous Assessment Activities

Title Weighting Hours ECTS Learning Outcomes
first exam 25 2 0.08 CM01, CM02, KM01, KM02, KM03, SM01, SM02, SM03
other activities 15 5 0.2 CM01, CM02, KM01, KM02, KM03, SM01, SM02, SM03
second exam 60 2 0.08 CM01, CM02, KM01, KM02, KM03, SM01, SM02, SM03

 
There will be a short test, which will provide a T grade. Some seminar sessions will be assessable. An S grade will be obtained from these tests. There will be a partial test at the end of the semester with a P1 grade. Based on these activities, a Final evaluation grade will be obtained, given by Final = 0.25 T + 0.15 S + 0.6 P1 If the Final grade is greater than or equal to 5, the student has passed the subject. Students who have not passed the subject may take a final retake exam where they can recover 85% of the grade. Single assessment. Students who have requested it can take advantage of the single assessment modality (see the Faculty's website). Single assessment implies an irrevocable waiver of the right to continuous assessment. The student who takes advantage of this assessment method will take three tests on the date of the last partial: an oral theory test, a written problem test and a written test corresponding to the contents of the seminars. The weight corresponding to each part is 25% for the theory part, 60% for the problems and 15% for the seminars. If the student does not pass the subject, he/she may opt for the retake exam under the same terms as the rest of the students.

Bibliography

M. Spivak. Calculus. Càlcul Infinitesimal. Ed. Reverté, Barcelona 1995.

L'assignatura de Funcions de Variable Real consisteix, essencialment, 

 

BIBLIOGRAFIA COMPLEMENTÀRIA

 

           . Hardy, G. H. A course of pure mathematics. Reprint of the (1952) tenth edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1992.

Un clàssic de veritat. Una mica antiquat en la notació i en alguns conceptes, però és la visió d'un matemàtic important. 

 

  • R. Larson, R. P. Hostetler, B. Edwards. Cálculo I. Ediciones Pirámide. 2002.

Un llibre amb un enfocament més pràctic. Conté nombrosos exemples, aplicacions i problemes.

  • J. M. Ortega. Introducció a l'Anàlisi Matemàtica. Manuals de la Universitat Autònoma de Barcelona 4, Bellaterra 1990. 

 El nostre curs consisteix en els cinc primers capítols d'aquest llibre. Aquest text  serà d'utilitat en alguns aspectes del curs com a complement de la referència bàsica.

  • W. Rudin. Principios de Análisis Matemático. Ed. McGraw-Hill. 1980.

Llibre de contingut més avançat que serà útil també en cursos posteriors.  Molt bona selecció de problemes.

  • R. Courant, H. Robbins. ¿Qué es la matemática? Aguilar, 1979. 

Segons una autoritat acadèmica actual: "Un clásico. Un libro de contenido más general que proporciona una magnífica visión global de la matemática."

  • G. Flory. Ejercicios de topología y de análisis. Tomos 1, 2.  Ed. Reverté, 1983. 

 Bons llibres de problemes de tipus més conceptual. Las parts que corresponen a aquest curs són el  capítol 1 del volum 1 i els capítols 5, 6, 7 del volum 2. Es tracta de llibres que seran útils també en cursos posteriors.

  •  B.P. Demidovich. 5000 problemas de Análisis Matemático. Paraninfo. 2000.

Llibre amb una completa selecció de problemes pràctics. Molt adient per exercitar conceptes i afiançar destresa de càlcul. 


Software

none


Groups and Languages

Please note that this information is provisional until 30 November 2025. You can check it through this link. To consult the language you will need to enter the CODE of the subject.

Name Group Language Semester Turn
(PAUL) Classroom practices 1 Catalan first semester morning-mixed
(PAUL) Classroom practices 2 Catalan first semester morning-mixed
(SEM) Seminars 1 Catalan first semester morning-mixed
(SEM) Seminars 2 Catalan first semester morning-mixed
(SEM) Seminars 3 Catalan first semester morning-mixed
(SEM) Seminars 4 Catalan first semester morning-mixed
(TE) Theory 1 Catalan first semester morning-mixed
(TE) Theory 2 Catalan first semester morning-mixed