Math 104 homework 6 determine the interior and closure of

Question

Math 104: Homework 6-

1. Determine the interior and closure of the following subsets of R:

A = {1/n: n ∈ N},               B = [0, 1] ∪ Q.

2. Consider the function

Show that h is continuous at 0 but at no other point.

3. In the lectures it was shown that a continuous map from [0, 1] to [0, 1] has a fixed point. Find an example of a continuous map from (0, 1) to (0, 1) which does not have a fixed point.

4. Let f be a real-valued function on R. Suppose that for a given x ∈ R,

lim_n→∞[ f(x + a_n) - f(x - a_n)] = 0

for all sequences (a_n) which converge to 0. Is f continuous at x?

5. Optional for the enthusiasts. A real-valued function f on an interval I is called convex if for all x, y ∈ I, and 0 < λ < 1, then

f((1 - λ)x + λy) ≥ (1 - λ)f(x) + λ f(y).

Suppose f is convex on [a, b]. Prove that f is continuous at x for a < x < b, but need not be continuous at a or b.