Math 3220
Introduction to Analysis
Summer term, 2000

Homework Assignments

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or to our grader/tutor Darrell Poore

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Homework originally due June 23 (extended to June 26)

HW1: Prove that f:R^n-->R^m is continuous if and only if f^{-1}(E) is closed for all closed subsets E in R^m.

HW2: Prove that the diamond region E:= {(x,y): y < x+1, y < -x+1, y > x-1, y > -x-1} is an open subset of R^2. Sketch the region.

HW3: Prove that the region E:={(x,y): x is greater than or equal to y^2, and x is less than or equal to 3} is closed.

HW4: Prove that f(x)=1/x is not uniformly continuous on the open interval (0,1).

HW5: (optional, assigned in Friday notes) For a less than or equal to b, and c less than or equal to d, sketch the closed rectangle E:=[a,b]x[c,d]:={(x,y)in R^2 such that x is in [a,b] and y is in [c,d]}. Prove that E is compact, and that E is path connected.

HW6: (optional, assigned in Friday notes) Prove that the union of the two balls B((-2,0),1) and B((2,0),1) is NOT path connected.

HW from the second edition: page 264, #1-3; pages 273-4, #2,5,8. For people working from the first edition, the text of these problems follows:

p. 264:
1) Identify which of the following sets are compact and which are not. If E is not compact, find the smallest compact set H (if there is one) such that E is a subset of H.
a) {1/k: k is a natural number} UNION {0}.
b) {(x,y) in R^2: x^2 + y^2 is greater than or equal to a and less than or equal to b}, where 0 < a < b are real numbers.
c) {(x,y) in R^2 : y=sin(1/x) for some x greater than zero and less than or equal to 1}.
d) {(x,y) in R^2: |xy| is less than or equal to 1}.

2) Let A and B be compact subsets of R^n. Prove that the union of A and B as well as the intersection of A and B are also compact. (You may prove this directly from the definition of compact, or via Arzela-Ascoli.)
3) Suppose E is a compact subset of the real numbers. Prove that sup(E) and inf(E) are elements of E.

pages 273-274:
2) Let f(x) be the square root of x, for x greater than or equal to 0. Let g(x)=1/x for nonzero x and define g(0)=0.
a) Find f(E) and g(E) for E=(0,1), E=[0,1), E=[0,1], and explain some of your answers by appealing to results in this section (which corresponds to section 5.6 in the first edition of Wade).
b) Find f^{-1}(E) and g^{-1}(E) for E=(-1,1) and E=[-1,1], and explain some of your answers by appealing to results in this section.

5) Prove that f(x,y):= exp(-1/|x-y|) when x is not equal to y, and f(x,y):=0 when x=y, is a continuous function on R^2.

8) This problem has been declared optional, and is now also a T-shirt reward problem (first 3 independent, correct solutions are eligible). It is now stated as follows:
Let E be a nonempty subset of R^n. Let f be a UNIFORMLY continuous function on E, to the image space R^m. Prove that there is a continuous extension function g defined on the closure of E, i.e. it agrees with f on E, i.e. f(x)=g(x) for all x in E.

Grading: A total of 24 points, divided as follows:
HW1: 2 points; HW2: 3 points (1 point for the sketch, 2 points to show the set is open); HW3: 2 points; HW4: 2 points; HW5-6: ungraded; page 264: 1abcd: 1 point each; 2: 2 points; 3: 2 points; page 273: 2a: 3 points (1 point for each E. You are only graded on whether you computed f(E), g(E) correctly. 2b: 2 points, same comments pertain as in part (a); 5: 2 points; 8: ungraded.