Math 446 Real Analysis I Fall 2011, Prof. Mark Janeba
|TTh 12:50 - 2:20
|Introduction to Real Analysis, 2nd edition, by
ISBN 978-0-321046-25-3, published by Addison Wesley
Course content:This course concerns itself with the concepts
of limit and continuity, which are the basis of mathematical analysis
and the calculus from which it evolved. We focus on limits of
sequences and of functions, and continuity of functions. While
the main context is the real numbers, i.e. limits of sequences of real
numbers, continuity of functions from the real numbers to the real
numbers, and so on, we will introduce a more general context, that of metric spaces,
in which to study the same concepts. This will allow us to extend
our results (often with no added effort) to functions on n, sequences in , functions on spaces of functions, etc. A highlight of the course is the interaction of continuity with the concepts of completeness, compactness, and connectedness,
which give us rigorous proofs of the seemingly obvious (but definitely
not obvious) existence of global extrema and Intermediate Value
Theorem. As time permits, we may begin the concept of
differentiability before semester break.
Note: In Real Analysis II, we will see that
the "proofs" about derivatives that we saw in Calculus I were mostly
quite rigorous, if we have a rigorous understanding of limits.
Therefore, we will quickly review what we already know about
derivatives in a rigorous setting. Integrals, however, are far
more complicated than a Calculus I/II course would lead us to believe,
especially if they involve discontinous functions. Thus much of
Real Analysis II will be concerned with properties of the Riemann
integral. Other material will include some introductory material
on measure theory and/or sequences of functions, e.g. uniform convergence.
At the completion of this course, the successful student will have demonstrated
- The ability to understand both abstract and concrete
- The ability to differentiate between sound mathematical
reasoning, flawed reasoning, and non-rigorous reasoning.
ability to use the basic tools and methods of proof seen in analysis,
in particular set theory and epsilon-delta and epsilon-n arguments.
- The ability to formulate and prove theorems that arise from the
definitions and concepts of the course content, and the ability to
apply those theorems to specific examples.
- The ability to write up, and occasionally present orally, one’s mathematical proofs and arguments in
clear and compelling manner.
|Three to five quizzes
|Two or three "midterm" exams:
|Two or three in class presentations
|Around 7 homework sets at ~20 pts each
For each graded piece of work, I will post cutoff scores for grades
of A-, B-, C, C-, and D. At the end of the term, if your point total is
more than the total of the A- cutoffs, your grade will be an A- or better,
and so on. Cutoffs will never be higher than: A-: 90%
B-: 80% C: 70% C-: 67%
D: 60% ... but they are often lower.
Exams: We will have a mix of
in-class and take-home exams. Exams will be announced several
days in advance. The "rules" for take-home exams will be provided
with the exam.
Final exam date: Freday Dec. 16, 2-5 p.m. For borderline grades, I tend
to pay more attention to the final exam score. The final may
be take-home, in which case the above is the due date.
Exam makeup policy: Midterm make-ups or early midterms are given
only for verifiable illness or for university-sanctioned intercollegiate
activities. For collegiate activities, you must see me before you
leave to arrange a makeup time. In any case, you must contact me in
except in emergencies.If the final is in-class, it will not be given
early; we will try to finalize this by mid-semester, but please plan
Quizzes are 10
to 25 minutes long. Some will be primarily content quizzes, e.g.
stating definitions and theorems from class and assigned reading; these
may be given without notice. Other quizzes will be short
reasoning and proof exercises, closely related to previously assigned
Presentations: I will
periodically assign individual presentations, typically proofs of
moderate difficulty, to be given during class meetings. For the
first presentation, I will ask students to give a preview in my office
so we can be confident that things are going well.
Homework is assigned daily but
sporadically. Approximately every other week, I will ask you to
turn in a specific subset of the assigned work. If you have
additional problems on the same page, it's fine if you turn them in as
well, but the problems specified for collection need to be thoroughly
legible and well written. If it is necessary to achieve
legibility and coherence, students may choose to (re)write up a clean
copy of their work to turn in (by the original deadline given).
In advanced mathematics, the step from things that can be easily
understood to things that may as well be spoken in the Martian language
is a short step. To stay ahead of this "step," one needs to keep
up with the concepts and definitions already covered. It is vital
that students do so. The occasional "pop" quiz on definitions and theorems
(see Quizzes above) and class discussion will
help motivate students to keep up and help me assess how well students
are doing. Please come prepared to discuss the
homework and assigned reading from the previous class.
Oops! (added 9/1) As
with all 400-level mathematics courses at Willamette, students of this
course are expected to attend the department colloquium at least four
times this semester. The purpose of this requirement is to
instill an appreciation of the breadth and variety of mathematical
topics, techniques, people and (academic) programs. If scheduling
conflicts make colloquium attendance impossible, alternative short
written assignments will be made. If a student anticipates
needing this alternative for more than one colloquium session, they
should contact the instructor well in advance so that suitable
assignments can be prepared. Failure to complete this requirement
will result in a modest grade penalty for the course.
Academic Honesty Expectations
All exams and quizzes are to be taken with books and notes closed (except
as noted on the exam paper), completely on your own. When we have take-home exams, specific expectations
will be distributed with those exams.
Plagiarism: any work copied or paraphrased from another source without
proper written acknowledgement. Plagiarized work will recieve a zero grade.
In keeping with college policy, plagiarism will be reported to the dean
(see student handbook). Systematic or organized plagiarism on exams or
quizzes will result in course failure. If you are uncertain about some
aspect of the academic honesty policy, it is your responsibility to get
clarification from the instructor.
Last Modified Sept. 1, 2011.
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