Continuity and Derivability of One Real Variable

With the development of the contents of this course (which are basic for other courses in the Degree) the student will acquire knowledge regarding some of the principal concepts, results and techniques which are used in the study of functions of one real variable, which is the main goal of Mathematical Analysis.
The achievement of these objectives will imply the knowledge of the theoretical contents of the course as well as being able to relate them and apply them to specific problems of different kinds, sometimes with computer aid. We will use the software Maxima or Maple to illustrate the concepts of the course.

0. Topological preliminaries.
Open and closed sets, accumulation points, compact and connected sets in R (quick overview of the topological contents of Introduction to Mathematical Analysis and Topology of Euclidean Spaces). (2h)
1. Limits
Limit of a function at a point. Left-hand and right-hand limit of a function at a point. Infinite limits and limits at infinity. Computation of limits: indeterminate limits. (5h)
2. Continuity
Continuity of a function at a point. Sequential continuity. Continuous functions: properties. Weierstrass and Bolzano Theorems. Continuity of monotone functions and their inverses. Uniform continuity. Heine’s Theorem. Continuous Extension Theorem. Sufficient and necessary criteria for uniform continuity. (8h)
3. Derivability
Derivative, left-hand and right-hand derivative of a function at a point. Geometrical and Physical interpretations of the derivative. Computation rules of derivatives. Local behavior of derivable functions: critical points. Darboux’s Theorem. Mean Value Theorem. Monotonicity and derivation. L´Hôpital’s Rule: application to the computation of indeterminate limits. (7h)
4. Higher order differentiation
Higher order derivatives. Concavity and convexity. Periodicity. Taylor Polynomial. Remainder formulas. Applications: approximate computations. Local study of a function. (6h)

In-library material, with reference:
Basic bibliography:
Bartle, R. G., Sherbert, D. R.. Introducción al Análisis Matemático de una variable. Limusa Wiley, 2010. (1202 196, 26 32)
Ballesteros, F. Ejercicios de análisis matemático. Autores 1994 (26 306)
de Burgos, J. Cálculo Infinitesimal de una variable, segunda edición. McGraw-Hill, 2007. (1202 381, 26 475, 26 424)

Complementary bibliography:
Ayres, F. Cálculo Diferencial e Integral. McGraw-Hill 1991 (1202 67)
Bradley, G. L. Cálculo de una variable. Prentice Hall 1998. (1202 318, 26 462)
Fernández Viña, J. A. Lecciones de Análisis Matemático I, Tecnos. (1202 17, 26 169)
Fernández Viña, J. A., Sánchez Mañes, E. Ejercicios y complementos de Análisis Matemático I, Tecnos. (1202 69)
Larson, R.E., Hostetler, R. P., Edwards, B. H. Cálculo. McGraw-Hill, 2006. (26 491)
M. Spivak. Cálculo infinitesimal. Reverté, 1994. (1202 95, 26 263)

On-line material:
• Aranda, Pepe. Cálculo infinitesimal en una variable. URL: http://www.iespppuquio.edu.pe/biblioteca/wp-content/uploads/2020/12/cal…
• Hardy, G. H. A Course of Pure Mathematics. Third Edition URL: https://www.gutenberg.org/files/38769/38769-pdf.pdf

During this course, the student will achieve, in different ways, all the competences gathered in the Plan of the Degree in Mathematics of the USC. In particular, the course will favor the acquisition of the following specific competences:
• To know the notions of limit, continuity, uniform continuity and differentiability of functions of one real variable.
• To express with precision and rigor, whether it is in oral or written form, the knowledge, procedures, results and ideas studied during the course.
• To identify errors in foul reasoning, proposing proofs or counterexamples.
• To recognize some of the problems for which the solving needs of the resources learned during the course (optimization problems etc.).
• To use the software Maxima or Maple as assistance in the development of those activities related to the contents of the course with the objective of improving concept understanding and the discovery and contrast of the results of the course.

In this and subsequent sections we will take into account the list of competences and references collected in the Plan of the Degree in Mathematics of the USC
http://www.usc.es/export9/sites/webinstitucional/gl/servizos/sxopra/mem…
In plenary lectures, we will develop the theoretical part of the course, illustrating it with examples in order to make it more comprehensible. Furthermore, we will reserve some time in order to solve exercises and sometimes we will pose questions in order to make students participate in a discussion. This dynamic is intended to work in competences CB1-CB5, CG1-CG5 and CE1-CE6.
In which respects the teaching in reduced groups, we intend to achieve more student participation, we will deal with problems and aspects of the course which are not treated during plenary lectures and we will analyze those matters of most difficult comprehension. In these sessions, we will work the competences CE7 and CE8.
Last, in lab lectures we will solve problems and, when they take place in the IT lecture rooms, we will deal with the computer software Maxima or Maple, in order to do calculations and graphic representations, which will be of use for the solution of problems and the understanding of the course material. In these lectures we will develop CE7-CE9 and CT1-CT3 and CT4.

All the training activities carried out in each group will be coordinated to guarantee the training equivalence of all the groups.

Attending to the specifications of the Memory of the Degree in Mathematicians of the USC, the students will have the opportunity to achieve some percentage of their final qualification by means of continuous assessment.

Continuous assessment (C)
The continuous assessment will consist in the realization of two intermediate tasks. These can be written works to be turned in during the course or written or on-line test that will take place during lecture time, on dates announced well in advance and coordinated with the rest of the course subjects. The activities proposed will be related with practical or theoretical aspects of the concepts of the subject, and might consist of individual or group tasks. Through these distinct activities, the professors will evaluate the acquisition of competences during the course. The qualification obtained in the continuous assessment will apply to the two opportunities of the academic course (second semester and July).

Final proof (E)
A final examination in the form of a written exam will be used to assess the knowledge achieved by the students in relation with the concepts and results of the subject and their capacity to apply them in specific cases, both from the theoretical and practical points of view, also valuing the clarity and the logical rigor shown. The competences acquired will be evaluated through this exam.

Although they will not necessarily be the same in the different groups, both the continuous assessment activities and the final exam will be coordinated to guarantee the formative equivalence of all the groups.

Calculation of the final grade: The numerical grade of the opportunity will be computed as max{E, 0’3C+0’7E} where E is the grade of the final exam of the opportunity (which will take place at the dates indicated by the Faculty) and C is the average of the continuous assessment.

Those students who do not participate at the final exam of a given opportunity will be scored as “not presented” in that opportunity.

In those cases of fraudulent behavior regarding assessments, the precepts gathered in the “Normativa de avaliación do rendemento académico dos estudantes e de revisión de cualificacións” will be applied.

IN LECTURE ROOM TIME
Plenary lectures (28 h)
Interactive classes in reduced groups (14 h)
Interactive classes of laboratory (14 h)
Reduced group tutoring (2 h)
Total: 58 h
PERSONAL WORK: About 92 h depending on the person and her background.

• To have a good knowledge of sequences of real numbers and real line topology.
• Daily study using bibliographical material, solving the proposed problems, summarizing the concepts of the course, repeating the definitions, etc.
• To plan beforehand the study time, keeping the study of the course up to date.
• To visit the lecturer’s office to consult any doubts concerning the course.