1. komponenta

Lecture type	Total
Lectures	60
Exercises	45

COURSE AIMS AND OBJECTIVES: The main objective of the course is to introduce first-year students to the foundations of mathematical analysis on the real line. More precisely, the course studies real numbers, sequences, convergence, and continuity. The teaching method combines rigorous proof and operational techniques through examples, encouraging the student to acquire both skills at the same time.

COURSE DESCRIPTION AND SYLLABUS:
1. Introduction. The sets N, Z, Q; properties of operations and order; representation on the number line; proof that the square root of 2 is not a rational number; motivation for R. The concept of a function; Cartesian coordinate system; graph of a real-valued function of a real variable; linear functions; fractional linear functions. Quadratic functions, polynomials, rational functions, composition of functions, injectivity, surjectivity, bijection, inverse function. Root functions, exponential function on Q, logarithmic function, hyperbolic and inverse hyperbolic functions. Trigonometric functions, inverse trigonometric functions, solving equations involving trigonometric functions. Axiomatic definition of the field R, supremum and infimum of a set, completeness. (6 weeks)
2. Sequences. Definition of a sequence and a subsequence, monotonicity, boundedness, monotone subsequence, various examples of sequences. Convergence, basic rules, relationship among convergence, boundedness, and monotonicity, Cauchy sequence, limit superior and limit inferior. The field C, sequences in C, convergence in C and componentwise. (3 weeks)
3. Continuity. Limit of a function and basic rules, continuity of a function and operations on continuous functions, continuity of rational functions. Rigorous introduction of the exponential function, continuity of the exponential function. Relationship among continuity, boundedness, and monotonicity, continuity of the inverse function. Continuity of root, logarithmic, hyperbolic, inverse hyperbolic, trigonometric, and inverse trigonometric functions. (4 weeks)

1. Matematička analiza 1 & 2, skripta, B. Guljaš.
2. Matematička analiza 1, S. Kurepa, Školska knjiga, Zagreb, 1997.
3. Mathematical Analysis 1, V. Zorich, Springer Verlag, Berlin, 2015.
4. Introduction to Real Analysis, R. G. Bartle, D. R. Sherber, John Wiley & Sons, 2011.
5. T. Tao, Analysis I, Hindustan Book Agency & Springer, 2022.

Mandatory course - Regular study - Mathematics