Test 2 Questions

Question 1

Let A be a nonempty set, and let f : A -> {0, 1} be a continuous, surjective function. (That is, there are points a ∈ A with f(a) = 0 and there are points b ∈ A
with f(b) = 1) Show that A is a disconnected set.

Answer 1

Define F as elements of A where f(x) = 0
The same for G where f(x) = 1
Clearly F∩G is empty and FUG = A

Let y be a limit point of F, then there is a sequence with (yn) -> y
Since f is continuous, f(yn) -> f(y), but since f(yn) = 0, f(y) = 0, and y ∈ F, f MUST be closed. Similarly G is closed

Thus F closure = F and the same for G, thus if F∩G is empty then they are separated, thus FUG represents a disconnected set, which is A.

Question 2

Let f : R -> R be continuous and let A = {x ∈ R : f(x)=0} Show that A is a closed
set.

Answer 2

Let y be a limit point of A, then there is a sequence yn with (yn) -> y. Since F is continuous, f(yn) -> f(y). Since both are zero, y must be in A, hence A is closed

Question 2

Let h(x) = 1/x^2
Show that h(x) is not uniformly continuous on (0,1)

Answer 2

Let xn = 1/x and yn = 1/x^2
We can use these because (xn) (yn) -> 0
Thus (xn - yn) -> 0

However, |h(xn) - h(yn)| = n^2|n^2 - 1|
If n >= 2, n^2 >= 4 and n^2 - 1 >= 3
Thus |h(xn) - h(yn)| >= 3n^2 which approaches infinity as n does, thus n is not uniformly continuous

Question 3

The complement of a connected open set is perfect.

Answer 3

True

Question 4

If f : (0, 1) ∈ R is continuous and {xn} ⇢ (0, 1) is Cauchy,
then {f(xn)} is Cauchy.

Answer 4

False