Reference no: EM131081545
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,
limn→∞[ f(x + an) - f(x - an)] = 0
for all sequences (an) 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.
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