MATH 3220 § 1 SIXTH HOMEWORK ASSIGNMENT Oct. 11, 2000 A. Treibergs Due Oct. 17, 2000

• Please hand in the following exercises from the text "Introduction to Analysis, Second Edition" by William Wade, Prentice Hall 1999.
• 273[ 2, 3, 6, 7 ].

• Here are the equivalent exercises for those using the First Edition:
• 256[ 7 ]
• Let f(x)=x1/2 and g(x)=1/x if x is nonzero and g(0)=0.
1. Find f(E) and g(E) for E=(0,1), E=[0,1) and E=[0,1], and explain some of your answers by appealing to the results in this section.
2. 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 the results in this section.
• Let f be a function taking Rn into Rm. Prove that the following are equivalent:
1. f is continuous.
2. f-1(V) is open in Rn for every open set V in Rm.
3. f-1(C) is closed in Rn for every closed set C in Rm.
• Since problem 273[6] is to prove the Intermediate Value Theorem which is done in the first edition, please do the alternate problem:
Let g be a function from [0,1] to R. Prove that if g takes on its values exactly twice, then g cannot be continuous at every point of [0,1]. [Bartle, Elements of Real Analysis, Wiley, 1964, p173. Bartle gives a hint.]