Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I: | (284) |
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University of Toronto, April 16, 2003

**Solve the following 6 problems. ** Each is worth 20 points
although they may have unequal difficulty, so the maximal possible
total grade is 120 points. Write your answers in the space below the
problems and on the front sides of the extra pages; use the back of the
pages for scratch paper. Only work appearing on the front side of
pages will be graded. Write your name and student number on each page.
If you need more paper please ask the presiding officers.

**Duration. ** You have 3 hours to write this exam.

**Allowed Material: ** Any calculating device that is not
capable of displaying text or graphs.

**Problem 1. ** We say that a set of real numbers is *dense* if for any open interval , the intersection is
non-empty.

- Give an example of a dense set whose complement is also dense.
- Give an example of a non-dense set whose complement is also not dense.
- Prove that if is a continuous function and for every in some dense set , then for every .

**Problem 2. ** Sketch the graph of the function
.
Make sure that your graph clearly indicates the following:

- The domain of definition of .
- The behaviour of near the points where it is not defined (if any) and as .
- The exact coordinates of the - and -intercepts and all minimas and maximas of .

**Problem 3. ** Compute the following integrals:

- .
- .
- ( ).
- .
- .

**Problem 4. ** Agents of the
CSIS have secretly developed two
functions, and , that have the following properties:

- and for all .
- and .
- and .

- and are everywhere differentiable and and .
- and .
- and for all .

**Problem 5. **

- Prove that if a sequence of integrable functions on an interval converges uniformly on that interval to a function , then the function is integrable on and .
- Prove that the series converges uniformly to some function on and write a series of numbers whose sum is .

**Problem 6. ** Let
be a sequence of complex
numbers and let be another complex number.

- Prove that the sequence is bounded iff the sequences and are both bounded.
- Prove that iff and .
- Prove that if the sequence is bounded then it has a convergent subsequence.

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Dror Bar-Natan 2003-04-28