Math 104: Homework 6 (due March 11)

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

A = {^{1}/n : n ∈ }, B = [0, 1] ∪ .

2. Consider the function

h(x) = \{ \begin{matrix}x & if x ∈ \\ 0 & if x ∉ . \end{matrix}

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

3. Ross exercise 17.9

4. 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.

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

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?

6. Ross exercise 18.9

7. Ross exercise 18.10

8. 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) + λf(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.