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Dror Bar-Natan

April 28, 2003

this document in PDF: Exam.pdf

**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. This booklet has
12 pages.

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

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

**Problem 1. ** Let and denote functions defined on
some set .

- Prove that
- Find an example for a pair , for which
- Find an example for a pair , for which

**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 a
function that has the following properties:

- for all .
- is differentiable at 0 and .

- is everywhere differentiable and .
- for all . The only lemma you may assume is that if a function satisfies for all then is a constant function.

**Problem 5. **

- Prove that if a sequence of continuous functions converges uniformly to a function on some interval , then is continuous on .
- Prove that the series converges on and that its sum is a continuous function of .

**Problem 6. ** Prove that the complex function
is everywhere continuous but nowhere differentiable.

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Dror Bar-Natan 2003-04-28