Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I: | (67) |
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**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.

(In the real exam each of the questions will appear on a separate page and there will be several blank pages stapled with your exam booklet. You will have space for your name and student number at the top of each page.)

**Problem 1. **

- Prove directly from the postulates for the real numbers and from the relevant definitions that if and then .
- Use induction to prove that any integer can be written in exactly one of the following three forms: , or , where is also an integer.
- Prove that there is no rational number such that .

**Problem 2. ** We say that a function is an *inverse* of a
function if
, where is the *identity
function,* defined by for all . Show that a function has an
inverse if and only if the following two conditions are satisfied:

- If then .
- For every there is an such that .

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

**Problem 4. ** Suppose that is, for each natural number
, some *finite* set of numbers and that and have no
members in common if . Define as follows:

**Problem 5. ** Suppose that satisfies
for all and and that is continuous at 0. Prove that is
continuous everywhere.

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Dror Bar-Natan 2002-10-23