PROBLEM SET IV
“BY EDWARD NASHTON”
DUE FRIDAY, 7 OCTOBER
Prove or give counterexamples to justify your claims.
Exercise 29. Is there a continuous function f : R
closed interval? Is there a continuous function f : R
open interval?
R such that the image of some closed interval is not also a
R such that the image of some open interval is not also an
R by the formula
j1
k u(s) := 2 s −
+s
.
2
Is u continuous? Consider the function v : R
R defined by the formula
(
u(1/t ) if t 6= 0;
v(t ) :=
0
if t = 0,
Exercise 30. Define the function u : R
and the function w : R
R defined by the formula
(
w(t ) :=
t u(1/t ) if t =
6 0;
0
if t = 0,
Is v continuous? Is w?
Definition. A subset E ⊂ R is said to be closed if its complement R − E is open.
R a continuous function. Must there be a function
Exercise 31. Suppose E ⊂ R a closed set, and suppose f : E
F:R
R such that for any x ∈ E one has F (x) = f (x)? What if E were only assumed open?
Exercise 32. Suppose E ⊂ R. Is the function dE : R
R defined by the formula
dE (x) := inf{|x − y| | y ∈ E}
continuous?
Exercise 33. Suppose E, E 0 ⊂ R two disjoint closed subsets. Must there be a continuous function f : R
that both f −1 (0) = E and f −1 (1) = E 0 ?
Exercise? 34. Let a < b be two real numbers. Is there a discontinuous function f : (a, b )
the sense that for any x, y ∈ (a, b ) and any t ∈ [0, 1], one has
R such
R that is convex, in
f ((1 − t )x + t y) ≤ (1 − t ) f (x) + t f (y)?
Exercise 35. Are there positive numbers r, s , t ∈ R with r ≥ t such that the limit
x
s + t 1/x
lim
x→0
r
does not exist? Are there positive numbers r, s, t ∈ R with r ≥ t such that this limit exists and is greater than 1?
1