# MAT-60206 Mathematical Analysis, 5 cr

INFO ABOUT THE EXAM ON APR 28, 2020: https://moodle.tuni.fi/course/view.php?id=1531 > News forum

This is the English version of the course MAT-60200 Matemaattinen analyysi. The course will be lectured in Finnish and in English in alternate years.
Suitable for postgraduate studies. The implementation will not be executed during the academic year 2019-2020.

Janne Kauhanen

#### Lessons

 Implementation Period Person responsible Requirements MAT-60206 2019-01 - Janne Kauhanen Final exam, weekly exercises, and weekly written exercises.

#### Learning Outcomes

This course introduces students to the fundamentals of mathematical analysis at an adequate level of rigor. Upon successful completion of the course, student will be able to: - Read mathematical texts and proofs, - Use the definitions and apply the basic results that are introduced during this course, and - Produce rigorous proofs of results that arise in this course using direct and indirect proof, induction and epsilon-delta technique.

#### Content

 Content Core content Complementary knowledge Specialist knowledge 1. THE REAL NUMBERS Field and ordering properties of the real numbers, supremum and infimum, the Completeness Axiom, limit points. Open and closed sets. Heine-Borel Theorem and Bolzano-Weierstrass Theorem. 2. THE LIMIT AND CONTINUITY OF FUNCTIONS DEFINED ON SUBSETS OF THE REAL LINE Continuity of the sum, the product, the quotient, and the composition of continuous functions. Boundedness, the existence of min and max and the Intermediate Value theorem for continuous functions. Uniform continuity. Monotone functions and continuity of the inverse function. 3. DIFFERENTIABILITY OF FUNCTIONS DEFINED ON SUBSETS OF THE REAL LINE Linear approximation, the derivative of the sum, the product and the quotient of differentiable functions, the Chain Rule, extreme values, the Mean Value Theorem, and l'Hospitals Rule. 4. THE RIEMANN INTEGRAL Riemann sums and upper and lower sums, existence of the integral using upper and lower sums (the Riemann Condition), integrability of monotone and continuous functions. Basic properties of the integral: linearity, monotonicity, integrability of the absolute value, and additivity on intervals. The Mean Value Theorem, the Fundamental Theorem of Calculus. Local integrability and the improper integral, improper integrals of nonnegative functions, the Comparison Test, absolute and conditional convergence. Integration by parts and the change of variable. 5. SEQUENCES AND SERIES OF FUNCTIONS Pointwise and uniform convergence of sequences of functions. Properties preserved by uniform convergence: continuity, differentiability, integrability. Pointwise and uniform convergence of series of functions.

#### Study material

 Type Name Author ISBN URL Additional information Examination material Book Introduction to real analysis (Edition 2.04) William Trench Chapters 1-4 No Summary of lectures Matemaattinen analyysi Janne Kauhanen Ladattavissa Moodlesta No

#### Prerequisites

 Course Mandatory/Advisable Description MAT-01360 Matematiikka 3 Mandatory 1 MAT-01366 Mathematics 3 Mandatory 1

1 . Matematiikka/Mathematics 1-3