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Assume that f is continuous everywhere. Show that for any non-empty subset UCR, if U is ajar, then f-¹(U) is ajar. Hint: To prove this, you need to prove and use these facts 1) for all non-empty A CR, AC f¹(f(A)). This proof should be one line proof; 2) f-¹(B₁) f¹(B₂) if non-empty sets B₁, B₂ satisfies B₁ C B₂ CR. This proof should be very short; and 3) the results from part c).

Assume that f is continuous everywhere. Show that for any non-empty subset UCR, if U is ajar, then f-¹(U) is ajar. Hint: To prove this, you need to prove and use these facts 1) for all non-empty A CR, AC f¹(f(A)). This proof should be one line proof; 2) f-¹(B₁) f¹(B₂) if non-empty sets B₁, B₂ satisfies B₁ C B₂ CR. This proof should be very short; and 3) the results from part c).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.1: Inverse Functions
Problem 18E
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Please answer part d of this question, thank you

For this problem, let f: R→ R be a function. We say that a non-empty set UCR is ajar if Vu € U, 3r> 0 such that (u-r, u + r) CU. (a) Show that (0, 2) is ajar. (b) Find a set that is not ajar. You don't need to prove it. My set is In the following parts, we will use the follow two definitions. For ACR, we define f(A) := {f(a): a € A}. For BCR, we define f¹(B) := {r € R: f(r) € B}. Note that f(A) and f-¹(B) are both sets. (c) Show that va € R, Ve > 0,36 > 0 such that f ((a-5, a +5)) (f(a)- e, f(a) + e) is equivalent to the definition of f is continuous everywhere.
Transcribed Image Text:For this problem, let f: R→ R be a function. We say that a non-empty set UCR is ajar if Vu € U, 3r> 0 such that (u-r, u + r) CU. (a) Show that (0, 2) is ajar. (b) Find a set that is not ajar. You don't need to prove it. My set is In the following parts, we will use the follow two definitions. For ACR, we define f(A) := {f(a): a € A}. For BCR, we define f¹(B) := {r € R: f(r) € B}. Note that f(A) and f-¹(B) are both sets. (c) Show that va € R, Ve > 0,36 > 0 such that f ((a-5, a +5)) (f(a)- e, f(a) + e) is equivalent to the definition of f is continuous everywhere.
Assume that f is continuous everywhere. Show that for any non-empty subset UCR, if U is ajar, then f-¹(U) is ajar. Hint: To prove this, you need to prove and use these facts 1) for all non-empty A CR, AC f¹(f(A)). This proof should be one line proof; 2) f-¹(B₁) Cf-¹(B₂) if non-empty sets B₁, B₂ satisfies B₁ C B₂ C R. This proof should be very short; and 3) the results from part c).
Transcribed Image Text:Assume that f is continuous everywhere. Show that for any non-empty subset UCR, if U is ajar, then f-¹(U) is ajar. Hint: To prove this, you need to prove and use these facts 1) for all non-empty A CR, AC f¹(f(A)). This proof should be one line proof; 2) f-¹(B₁) Cf-¹(B₂) if non-empty sets B₁, B₂ satisfies B₁ C B₂ C R. This proof should be very short; and 3) the results from part c).
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