QuestionQuestion

Section 11.3

2. Let f: R --> R be defined by f(x) := 1/(1 + x²) for x ∈ R.
(a) Find an open interval (a,b) whose direct image under f is not open.
(b) Show that the direct image of the closed interval [0, ∞) is not closed.

4. Let h: R --> R be defined h(x) := 1 if 0 ≤ x ≤ 1, h(x) := 0 otherwise. Find an open set G such that h⁻¹(G) is not open, and a closed set F such that h⁻¹(F) is not closed.

8. Give an example of a function f: R --> R such that the set {x ∈ R: f(x) = 1} is neither open nor closed in R.

Section 11.4

1. Show that the functions d₁ and d∞ are metrics on R².

2. Show that the functions d∞ and d₁ are metrics on C[0,1].

8. Let P := (x,y) and O := (0,0) in R². Draw the following sets in the plane:
(a) {P ∈ R² : d₁(O, P) ≤ 1}.
(b) {P ∈ R² : d∞(O,P) ≤ 1}.

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