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)=x^{1/2} and g(x)=1/x if x is nonzero and
g(0)=0.
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.
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 R^{n} into R^{m}. Prove that the following are
equivalent:
f is continuous.
f^{-1}(V) is open
in R^{n} for every open set V in
R^{m}.
f^{-1}(C) is closed in
R^{n} for every closed set C in
R^{m}.
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.]