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.

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.

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.