Introduction to Mathematical Analysis

Introduce students, with essential support from examples and practice, to the understanding of the first structure of Mathematical Analysis, the ordered and complete field of real numbers, and the fundamentals of real valued real functions.

Introduce and consolidate, through examples and exercises, the notions of convergence of sequences and numerical series.

Present, with practice using different notations, operations with complex numbers.

Introduce the different notions of limits of real valued real functions, and study continuity and uniform continuity of these functions.

1. REAL NUMBERS (approx. 8 lecture classes)
1.1 Natural numbers. Principle of induction.
1.2 Rational numbers. Countability.
1.3 Axiomatic structure of the real numbers (R). Supremum axiom and consequences.
1.4 Archimedean property of R. Density of Q in R.

2. SEQUENCES OF REAL NUMBERS (approx. 9 lecture classes)
2.1 Intuitive introduction to the concepts of sequence and limit. General notions.
2.2 Convergent sequences and their limits. Properties.
2.3 Infinite limits.
2.4 Convergence and divergence of monotonic sequences.
2.5 Subsequences. Bolzano-Weierstrass Theorem. Oscillation limits.
2.6 Cauchy sequences. Completeness of ℝ.
2.7 Limit calculations. Stirling’s and Stolz’s criteria.

3. SERIES OF REAL NUMBERS (approx. 8 lecture classes)
3.1 Intuitive introduction to the concept of series and its sum.
3.2 Numerical series. Convergence of series.
3.3 Series with non-negative terms. Convergence criteria.
3.4 Absolute and conditional convergence. Non-absolute convergence criteria.
3.5 Decimal expression in R and other numeral systems.

4. COMPLEX NUMBERS (approx. 2 lecture classes)
4.1 Complex numbers. Binomial form and basic operations.
4.2 Exponential form and its consequences: powers, roots, Euler’s and De Moivre’s formulas.

5. LIMITS (approx. 7 lecture classes)
5.1 Topological preliminaries in R.
5.2 Limit of a function at a point.
5.3 Lateral limits.
5.4 Infinite limits and limits at infinity.
5.5 Calculating limits: Indeterminate forms.

6. CONTINUITY (approx. 8 lecture classes)
6.1 Continuity of a function at a point.
6.2 Sequential continuity.
6.3 Continuous functions: Properties.
6.4 Weierstrass and Bolzano theorems.
6.5 Continuity of monotonic functions and their inverses.
6.6 Uniform continuity.
6.7 Heine’s Theorem.
6.8 Continuous extension theorem.
6.9 Sufficient and necessary criteria for uniform continuity.

BÁSICA
 
[1] T.M. Apostol. Análisis Matemático (2ª Ed.). Reverté, 1979.
 
[2] R.G. Bartle, D.R. Sherbert. Introducción al Análisis Matemático de una Variable (3ª Ed.). Limusa Wiley, 2010.
 
[3] R. Figueroa Sestelo, Ó.A. Otero Zarraquiños. Números reais. Universidade de Santiago de Compostela, 2022. Available online at: https://iacobus.usc.gal/permalink/34CISUG_USC/o7pcup/alma99101373073540…
 
[4] R. Figueroa Sestelo, Ó.A. Otero Zarraquiños. Números complexos. Universidade de Santiago de Compostela, 2022. Available online at: https://iacobus.usc.gal/permalink/34CISUG_USC/1sh7577/alma9910135587473…
 
[5] R. Figueroa Sestelo, Ó.A. Otero Zarraquiños. Series de números reais. Universidade de Santiago de Compostela, 2022. Available online at: https://iacobus.usc.gal/permalink/34CISUG_USC/1sh7577/alma9910135587472…
 
[6] R. Figueroa Sestelo, Ó.A. Otero Zarraquiños. Sucesións de números reais. Universidade de Santiago de Compostela, 2022. Available online at:
https://iacobus.usc.gal/permalink/34CISUG_USC/1sh7577/alma9910135587471…
 
[7] F. A. F. Tojo. Introducción al estudio de funciones de una variable real. Universidade de Santiago de Compostela, 2022.
Available online at: https://iacobus.usc.gal/permalink/34CISUG_USC/o7pcup/alma99101373082020…
 
 
COMPLEMENTARIA
 
[1] T. Gowers, J. Barrow-Green, I. Leader. The Princeton Companion to Mathematics. Princeton University Press. 2008. Available online at: https://iacobus.usc.gal/permalink/34CISUG_USC/o7pcup/alma99101352295090…
 
[2] T. Tao. Analysis I. Second Edition, Hindustan Book Agency.  Available online at: https://iacobus.usc.gal/permalink/34CISUG_USC/o7pcup/alma99101346685280…
 
[3] S. Behar Jequín, R. Roldán Inguanzo, A. Arredondo Soto. Análisis matemático real: ejercicios y problemas. Universidad de La Habana, 2021.
Available online at: https://elibro-net.ezbusc.usc.gal/es/lc/busc/titulos/196988
 

As stated in the current Degree Report of the Bachelor's Degree in Mathematics at USC, the following are the knowledge, skills, and abilities to be developed:

KNOWLEDGE

Con01: Understand key concepts, methods, applications, and results across the main branches of Mathematics.
Con02: Understand and use mathematical language to construct and interpret proofs and formulate models.
Con03: Know the proofs of relevant theorems in various mathematical branches.
Con04: Internalize the definition of mathematical objects, relate them to others, and apply them in different contexts.
Con05: Abstract essential properties and facts of a problem and identify appropriate mathematical tools to address it.

COMPETENCIES

Comp01: Gather and interpret relevant data and results to draw conclusions and issue reasoned reports on scientific, technological, or other problems requiring mathematical tools.
Comp02: Communicate mathematical knowledge, procedures, results, and ideas clearly, both orally and in writing, to both specialized and general audiences.
Comp03: Learn new knowledge and techniques from various branches of mathematics.

SKILLS / ABILITIES

H/D01: Apply theoretical and practical knowledge, along with analytical and abstract reasoning skills, to define and solve problems in academic or professional contexts.
H/D02: Use general and specialized mathematical bibliographic resources and search tools.
H/D03: Organize and plan work effectively.
H/D04: Verify arguments and reasoning, identify errors, and suggest revisions or counterexamples.
H/D06: Read scientific texts in the native language and in other languages of relevance to the scientific field.
H/D07: Develop mathematical proofs, formulate conjectures, and devise strategies to confirm or refute them.

The general methodological guidelines established in the current Degree Report of the Bachelor's Degree in Mathematics at USC will be followed.

Lecture classes will mainly consist of instructor-led sessions dedicated to explaining theoretical content and solving selected problems or exercises. These may vary between a traditional lecture style and a more participative format. They will primarily support the acquisition of knowledge Con01 to Con05 and also contribute to competency Comp02 and skill H/D07.

Interactive classes will focus on practical skills development and immediate application of theoretical concepts through problems and exercises. These will emphasize H/D01, H/D04, and H/D07, as well as competencies Comp01 and Comp02, while reinforcing the foundational knowledge.

Tutorials will provide individual or small-group support for addressing specific questions related to assignments or understanding the material.

All student tasks (study, readings, exercises, practicals, etc.) will be guided by instructors through interactive classes and small-group tutorials, supporting skills H/D02, H/D03, and H/D06 and competencies Comp01 to Comp03.

A virtual learning platform will be used to provide essential course resources (explanatory videos, notes, exercise booklets, etc.) and to facilitate official communication between instructors and students.

Assessment combines continuous evaluation and a final exam, coordinated across all lecture groups, to measure the acquisition of knowledge, competencies, and skills.

FINAL EXAM (FE)

A single written exam for all groups, graded out of 10 points, will assess theoretical understanding, reasoning ability, and application of mathematical techniques. The exam content is weighted approximately as follows (each part ~25%):

-Statements and proofs of lemmas, propositions, theorems, and corollaries: evaluates theoretical knowledge (Con01, Con03), formal language use (Con02), and proof skills (H/D07).

-Theoretical questions: assesses conceptual understanding, argumentation, and relationships between concepts (Con02, Con04, H/D04).

-Practical questions: tests the use of specific techniques (Con05, H/D01).

-Problems or exercises: measures problem-solving ability and clarity of presentation (Comp01, H/D01, H/D03).


CONTINUOUS ASSESSMENT (CA)

Continuous assessment consists of two in-person midterm tests, each graded out of 10. These tests progressively cover course content and evaluate both knowledge acquisition and skill development. Each test is weighted approximately as follows (each part ~33%):

-Theoretical questions: assess progressive assimilation of key concepts and correct use of mathematical language (Con01, Con02, Con04).

-Practical questions: measure the ability to apply techniques to specific problems (Con05, H/D01, H/D04).

-Problems or exercises: evaluate ability to tackle new situations, justify solutions, and communicate clearly (Comp01, Comp02, H/D01, H/D03, H/D07).

The continuous assessment grade is calculated as:
CA = (1/2)*P1+(1/2)*P2
where P1 and P2 are the scores of the two tests.

Although structure and activity types are the same for all lecture groups, content may be adapted to each group, ensuring coordination and equivalency.


FINAL GRADE (FG)

The final grade is calculated as:

FG = max {FE, 0.3 × CA + 0.7 × FE}

This formula values consistent work and progress while ensuring fair and rigorous final evaluation.

Students who do not attend the final exam will be marked as Not Presented.


SECOND EXAM PERIOD
The second exam follows the same structure. The continuous assessment score from the first attempt will still apply. The final exam will be common to all groups and evaluated according to the same criteria.


ADDITIONAL CONSIDERATIONS
In cases of academic misconduct (e.g., plagiarism or improper use of technology), the university's academic performance and grading review regulations will be enforced.

CLASSROOM ACTIVITIES
Lecture classes (42 hours)
Interactive lab classes (42 hours)
Small-group tutorials (2 hours)
Total in-class hours: 86

STUDENT'S PERSONAL WORK
Independent or group study (77 hours)
Writing exercises, conclusions, or other assignments (40 hours)
Programming/experimentation or other computer/lab work (12 hours)
Recommended readings, library activities, or similar (10 hours)
Total hours of personal work: 139