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University of Toronto, November 29, 2004

This document in PDF: Exam.pdf

**Solve all of the following 5 problems. **
Each problem is worth 20 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 tutors. You have an
hour and 50 minutes.

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

**Problem 1. ** Let and be continuous functions defined
on all of
.

- Prove that if for some , then there is a number such that whenever .
- Prove that if two continuous functions are equal over the rationals then they are always equal. That is, if for every then for all .

**Problem 2. ** Let be a continuous function defined on
all of
, and assume that is rational for every
. Prove that is a constant function.

**Problem 3. ** We say that a function is *locally
bounded* on some interval if for every there is an
so that is bounded on
.
Let be a locally bounded function on the interval and let
is bounded on and .

- Justify the definition of : How do we know that exists?
- Prove that .
- Prove that .
- Prove that .
- Can you summarize these results with one catchy phrase?

**Problem 4. **

- Define `` is differentiable at ''.
- Prove that if is differentiable at then it is also continuous at .

**Problem 5. ** Draw an approximate graph of the function
making sure to clearly indicate
(along with clear justifications) the domain of definition of , its
-intercepts and its -intercepts (if any), the behaviour of at
and near points at which is undefined (if any),
intervals on which is increasing/decreasing, its local
minima/maxima (if any) and intervals on which is convex/concave.

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Dror Bar-Natan 2004-11-25