Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I: | (68) |
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University of Toronto, October 21, 2002.

this document in PDF: Exam.pdf

**Solve the following 5 problems. ** Each is worth 20 points
although they may have unequal difficulty. 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 tutors. You have an
hour and 50 minutes.

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

**Problem 1. **

- Prove directly from the postulates for the real numbers and from the relevant definitions that if and , then . If you plan to use a formula such as you don't need to prove it, but of course you have to be very clear about how it is used.
- Use induction to prove that any integer can be written in exactly one of the following two forms: or , where is also an integer.
- Prove that there is no rational number such that .

**Problem 2. **

- Suppose . Are there any functions such that ?
- Suppose that is a constant function. For which functions does ?
- Suppose that
for
*all*functions . Show that is the identity function .

**Problem 3. ** Sketch, to the best of your understanding, the
graph of the function

**Problem 4. ** Write the definition of
and give examples to show that the
following definitions of
do not
agree with the standard one:

- For all there is an such that if , then .
- For all there is a such that if , then .

**Problem 5. ** Suppose that is continuous at 0 and
and that
for all . Show that is
continuous at 0.

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Dror Bar-Natan 2002-10-23