PROBLEM SET III
“BY EDWARD NASHTON”
DUE FRIDAY,  SEPTEMBER 
Prove or give counterexamples to justify your claims.
Exercise . Is there a continuous function f : R . R such that the image of
some closed interval is not also a closed interval? Is there a continuous function
f : R . R such that the image of some open interval is not also an open interval?
Exercise . Define the function u : R . R by the formula
(
⌊1
⌋)
.
u(s) := 2 s −
+s
2
Is u continuous? Consider the function v : R . R defined by the formula
{
u(1/t) if t ̸= 0;
v(t) :=
0
if t = 0,
and the function w : R .
R defined by the formula
{
tu(1/t) if t ̸= 0;
w(t) :=
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.
Exercise . Suppose E ⊂ R a closed set, and suppose f : E . R a continuous
function. Must there be a continuous function F : R . R such that for any x ∈ E
one has F(x) = f(x)? What if E were only assumed open?
Exercise . Suppose E ⊂ R. Is the function dE : R .
R defined by the formula
dE (x) := inf{|x − y| | y ∈ E}
continuous?
Exercise . Suppose E, E′ ⊂ R two disjoint closed subsets. Must there be a
continuous function f : R . R such that both f−1 (0) = E and f−1 (1) = E′ ?
Exercise⋆ . Let a < b be two real numbers. Is there a discontinuous function
f : (a, b) . R that is conex, in the sense that for any x, y ∈ (a, b) and any t ∈
[0, 1], one has
f((1 − t)x + ty) ≤ (1 − t)f(x) + tf(y)?


DUE FRIDAY,  SEPTEMBER 
Exercise . Define a function θ : R . [0, 1] by the formula
{
0
if x ∈
/ Q;
θ(x) :=
1/q if x = p/q, where p, q ∈ Z, gcd(p, q) = 1 and q ≥ 1.
At which numbers x ∈ [0, 1] is θ continuous?