Various problems that are out of standard course of mathematical analysis are discussed during the course. Problems are gathered in cycles that start from simple, standard problems and finish with non-trivial mathematical statements.
The course includes short lectures or reviews, students work and group discussions.
The course may be useful to prepare for mathematical competitions.
Examples of few groups of problems that may be given in a course (the first problem is simple, the last is advanced):
Topic: Numbers. Problem. (a) Prove that (2)^1/2??, (b) Prove that (2)^1/2+ (3)^1/2??, (c) Prove that (2)^1/2+ (3)^1/2+ (5)^1/2??, (d) Prove that (2)^1/2+ (3)^1/2+ (5)^1/3??, (e) e=2.718...??, ...
Topic: Continuity. Problem. (a) Find a function that is discontinuous at 0. (b) Find a function that is discontinuous at two points. (c) Find a function that is discontinuous at countable number of points. (d) Find a function that is discontinuous at countable number of points that lies in [0,1]. (e) Let f(x)=1/n, if x=m/n is in lowest terms and f(x)=0 otherwise. Prove that f is discontinuous at any rational point and continuous at any irrational number. (f) Find an increasing function that is discontinuous at any rational point and continuous at any irrational number.
Tentative list of topics:
1) Sets (countable, uncountable, etc.);
2) Real number system (rational, irrational numbers, ...),;
3) Sequences (boundedness, supremums, infimums, limits);
4) Continuity;
5) Derivative;
6) Integral.
Advised prerequisites: Basic course of Mathematical Analysis