SUBJECT

Title

Analysis 1

Type of instruction

lecture + practical

Level

bachelor

Faculty

Faculty of Informatics

Part of degree program

Credits

3+2

Recommended in

Semester 2

Typically offered in

Spring semester

Course description

The set of real numbers, bounded sets, least upper bound (sup), greatest lower bound (inf).
Numerical sequences, monotone sequences. Convergence, Cauchy criterion.
Algebraic operations and convergence.
Convergence of monotone sequences.
The n-th root of numbers.
Extended real number line, limit in extended sense. Infinite numerical series, convergence, absolute convergence. Convergence tests. Alternating (Leibniz type) series.
The associativity (brackets in the series).
The permutation of the terms.
Products of series, Mertens’ theorem. padic fraction representation of real numbers.
Power series, Cauchy-Hadamard theorem.
Sum function of power series, elementary functions.
Limits of functions.
„Transfer principle”, limits and algebraic operations.
Limits of analytic and monotone functions. Continuity, discontinuity.
Connections between limit and continuity.
„Transfer principle” for continuous functions, algebraic operations with continuous functions.
Continuity of composition of functions.
Bolzano’s theorem, Darboux property.
Continuity of analytic functions.
Extremal values of continuous functions on compact intervals.
Weierstrass’ theorem.
niform continuity, Heine’s theorem.
Continuity of inverse functions.

Readings

T. Tao: Analysis I (Hindustan Book Agency (India), 2006)
G. B. Thomas - M. D. Weir - J. Hass - F. R. Giordano: Thomas's Calculus, 11th Ed. (Pearson Education, Inc, 2005)
Leindler László, Schipp Ferenc: Analízis I. (egyetemi jegyzet, Tankönyvkiadó, Budapest, 1976)
Pál Jenő, Schipp Ferenc, Simon Péter: Analízis II. (egyetemi jegyzet, Tankönyvkiadó,Budapest, 1982)
Szili László: Analízis feladatokban I. (ELTE Eötvös Kiadó, Budapest, 2008)

Recommended literature:

Balázs M., Kolumbán J.: Matematikai analízis (Dacia Könyvkiadó, Kolozsvár-Napoca, 1978)
Schipp Ferenc: Analízis I. (egyetemi jegyzet, JATE, Pécs, 1994)
Simon Péter: Fejezetek az analízisből (egyetemi jegyzet, ELTE Természettudományi Kar,Budapest, 1997)
W. Rudin: A matematikai analízis alapjai (Műszaki Könyvkiadó, Budapest, 1978)