787.03 Summer 2003 Week 6
Week 6 (week of July 28). Topic for the week:
Intermediate value property, uniform continuity, continuous functions
in metric spaces, Derivative of a real function, Mean Value Theorems
(Kaczor and Nowak V.II sections 1.3, 1.5, 1.7, 2.1, 2.2; Berkeley Problems sections 1.2, 1.4).
On Monday, July 28, we finished the proof that a real function with
only simple discontinuities has only countably many
discontinuities by handling the case of a removable
discontinuity. We solved problem 1.2.12 in the Berkeley book,
and talked about problems 1.3.3, 1.3.6, 1.3.20, and 1.3.16 in
Kaczor and Nowak, V.II. I started problem 1.3.28 (will finish
next time). After class was over I noticed I mistated the
problem: it should say f: R -> [0, \infty), not the other way
around.
On Tuesday, July 29, we finished problem 1.3.28 (with a little help
from the audience!) I suggested we look at problems 1.5.7,
1.5.9, 1.5.11, 1.5.13, 1.5.18, and 1.5.21 from Kaczor and Nowak
V.II; Problems 1.2.6 and 1.2.9 from the Berkeley book; and the
following problems from old qualifying exams: Spring 00 #1, Sp
99 #5. Also interesting (dealing with discontinuities) were Au
94 #3, Sp 94 #2, Sp 92 #4.
Then we worked out problems 1.5.12 and 1.5.13 from Kaczor
and Nowak, V.II.
On Wednesday, July 30, we did problem #1 from Spring 00
qualifying exam, and a problem on upper semicontinuity (Berkeley
1.2.9). I reviewed the Baire Theorem and used it for Kaczor and Nowak
V.2, problem 1.7.23.
On Friday, August 1, we had a new homework assignment: do the
Spring 2002 qualifying exam, and Berekely problem 2.4.27. I did
Kaczor and Nowak problem 1.7.22 (a more difficult problem using the
Baire Theorem), Berkeley 1.2.4 and 1.2.11, and Kaczor and Nowak
1.7.10.
Matthew Stenzel
Last modified: Mon Aug 4 12:20:42 EDT 2003