PROBLEM SET IV
“WORDY”
DUE FRIDAY,  OCTOBER 
Exercise . Suppose
I1 ⊃ I2 ⊃ · · · a sequence of closed intervals. Show that
∩
the intersection ∞
I
k=1 k ̸= ∅.
√
Exercise . Show that for any odd natural number n, the n-th root n x exists for
any x ∈ R. at is, show that there exists y ∈ R such that yn = x. Show that the
same holds for any nonnegative x if n is even.
Exercise . Use the previous problem to define, for every rational number s ∈
Q, a new function rs √
: (0, +∞) . R such that if s = p/q in lowest common
terms, then rs (x) = ( q x)p . Is rs continuous?
Definition. Suppose E ⊂ R. en a map f : E . R is uniformly continuous if for
any ε > 0, there exists a quantity δ > 0 such that for any points x0 , x1 ∈ E, if
|x0 − x1 | < δ, then | f (x0 ) − f (x1 )| < ε.
Exercise . Suppose E ⊂ R. Show that a uniformly continuous map E . R
is continuous, and show that the converse holds if E is a union of finitely many
closed intervals. Nevertheless, give an example of a continuous function f : R . R
that is not uniformly continuous.
Exercise⋆ . Show that no continuous function f : R . R takes every value exactly twice. Nevertheless, find a continuous function f : R . R takes every value
exactly three times. (Given a natural number n, when do you suppose that there
is a continuous function f : R . R that takes every value exactly n times?)
Definition. Suppose f : R . R a continuous function. A shadow point for f is a
real number x such that for some real number y < x, one has f (y) > f (x). (Draw
a picture of this to get the idea.)
Exercise . Suppose f : R . R a continuous function, and suppose that a ≤ b
are real numbers. Assume that a and b are not shadow points for f and that if every
c ∈ (a, b) is a shadow point for f. Prove that in fact f (a) = f (b).
Definition. Suppose E ⊆ R. A subset S ⊆ E is said to be dense in E if, for every
open subset U ⊂ R, if U ∩ E = ∅, then U ∩ S = ∅.
Exercise . Prove that both Q and R−Q are dense in R. Show that for any finite
set of points x1 , . . . , xn in an interval I, the set I − {x1 , . . . , xn } is dense in I.
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DUE FRIDAY,  OCTOBER 
Exercise . Suppose E ⊂ R, and suppose S ⊂ E dense in E. Suppose that
f : E . R and g : E . R are two continuous functions such that for any x ∈ S,
one has f (x) ≤ g(x). Show that for any x ∈ E, one has f (x) ≤ g(x). If one assumes instead that for any x ∈ S, one has f (x) < g(x), does it follows that for any
x ∈ E, one has f (x) < g(x)? Prove or give a counterexample.