Math 3220
Introduction to Analysis
Summer term, 2000

Homework Assignments

Send e-mail to : Professor Korevaar
or to our grader/tutor Darrell Poore

Links:
Math 3220 homework page
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Homework due Friday June 16

% 9.1 (2nd Edition), pages 259-261: 1cd, 2abc, 3(prove!), 7,8.
In the 1st Edition these correspond to p.231 #2abc, 3,9,10, except that 1cd aren't there. These are the problems:
1c) find interior, closure, boundary for E equal to the union of the intervals (1/(n+1),1/n) in the reals, as n ranges in the natural numbers.
1d) find the interior, closure and boundary for E equal to the union of the intervals (-n,n), as n ranges in the natural numbers.

HW problem I: The first three independent solutions of this problem win Math Department T-shirts! (This problem is optional.) It has two parts:
Ia) Find a subset E in the reals so that repeatedly using the operations of interior and closure creates seven distinct sets (including E).
Ib) Prove that for every subset E in R^n, the interior and closure operations can never create more than seven distinct sets.
Yes! We have a winner, submitted on June 16th. But two more people can still win...

HW problem II:
IIa) Prove that the quotient of two continuous real-valued functions is continuous, provided the denominator never equals zero. (This amounts to recalling what continous means, and then quoting the appropriate limit theorem.)
IIb) Prove that f:R^n-->R^m is continuous if and only if each of the component functions f_i(x) is continuous, for all i=1,2,...m.
IIc) Prove that the coordinate functions f(x_1, x_2, ...x_n)=x_j are continuous, for all j=1,2,...n.

Grading: 19 points, distributed as follows:
1cd: 1 point each; 2abc: 2 points each; 3: 2 points; 7abc: 1 point each; 8: 2 points; HW IIac: 1 point each; HW IIb: 2 points.