# Exam

The exam is posted online on **Feb 1, 2016**, and must be submitted on the last day of the exams’ period, **February 26**. Its goals are, besides testing your acquired skills in the Analysis, to teach you a few extra things and see your ability for logical reasoning, not your proficiency in performing long computations. If you find yourself involved in heavy computations, better double check whether you understand the formulation of the problem correctly. Remember, small details sometimes matter!

Please provide argumentation, better in the form of logical formulas, not forgetting explicit or implicit quantifiers and . They really may change the meaning of what you write!

Problems are often subdivided into items. The order of these items is not accidental, try to solve them from the first till the last, and not in a random order (solution of one item may be a building block for the next one).

To get the maximal grade, it is *not necessary* to solve all problems, but it is imperative *not to write* stupid things. Please don’t try to shoot in the air.

The English version is the authoritative source, but if somebody translates it into Hebrew (for the sake of the rest of you) and send me the translation, I will post it for your convenience, but responsibility will be largely with the translator.

If you believe you found an error or crucial omission in the formulation of a problem, please write me. If this will be indeed the case (*errare humanum est*), the problem will be either edited (in case of minor omissions) or cancelled (on my account).

That’s all, folks!© Good luck to everybody!

**Yes, and feel free to leave your questions/talkbacks here, whether addressed to Michal/Boaz/me or to yourself, if you feel you want to ask a relevant question. **

# Corrections

## Correction 1

The formulation of Problem 1 was indeed incorrect. The set was intended to be the set of accumulation points for a set . The formal definition is as follows.

*Definition*. A point belongs to to the set of limit points if and only if >0 the intersection is **infinite**. The point itself may be or may be not in .

Isolated points of are never in , but may contain points .

Apologies for the hasty formulation.

## Correction 2: Problem 3(b) cancelled!

**The statement requested to prove in Problem 3(b) is wrong**, and I am impressed how fast did you discover that. Actually, the problem was taken from the textbook by Zorich, vol. 1, where it appears on p. 169, sec. 4.2.3, as Problem 4.

The assertion about existence of the common fixed point of two commuting continuous functions becomes true if we require these functions to be continuously differentiable on (in particular, for polynomials), but the proof of this fact is too difficult to be suggested as a problem for the exam.

Thus **Problem 3(b) is cancelled.**