Math 205a Fall Term 2002

Assignment 5

1. Let C_{1} and C_{2} be compact subsets of ℝ. Show that

C_{1} + C_{2} = {x + y | x ∈ C_{1} and y ∈ C_{2}}

is also compact.

2. Give examples of a continuous function which is not Hölder continuous of any exponent and of a Hölder continuous function which is not Lipschitz.

3. Let f ∈ C(ℝ). Then [f = α] = {x ∈ ℝ | f(x) = α} is closed ∀α ∈ ℝ

[f > α] = {x ∈ ℝ | f(x) > α} is open ∀α ∈ ℝ

Show that

∂[f > α] ⊂ [f = α] .

4. Let g, h ∈ C(ℝ) and prove that f : ℝ → ℝ given by

f(x) := \begin{cases}g(x) , & x ≤ a \\ h(x) , & x > a \end{cases}

is continuous iff g(a) = lim_{x→a+} h(x).

The Homework is due Tuesday, November 12