# s3 - Math 8601 REAL ANALYSIS Fall 2010 Homework#3 Problems...

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Unformatted text preview: Math 8601: REAL ANALYSIS. Fall 2010 Homework #3. Problems and Solutions. #1. Let f be a real function on R 1 . The image and the inverse image of a subset A ⊂ R 1 under f are correspondingly f ( A ) = { y : y = f ( x ) for some x ∈ A } , f- 1 ( A ) = { x : f ( x ) ∈ A } . Show that f ( f- 1 ( A )) ⊂ A ⊂ f- 1 ( f ( A )) for arbitrary A ⊂ R 1 . Give an example when f- 1 ( f ( A )) 6 = A . Solution. (i). y ∈ f ( f- 1 ( A )) = ⇒ y = f ( x ) for some x ∈ f- 1 ( A ) = ⇒ y = f ( x ) ∈ A . Therefore, f ( f- 1 ( A )) ⊂ A . (ii). x ∈ A = ⇒ y = f ( x ) ∈ f ( A ) = ⇒ x ∈ f- 1 ( f ( A )). Therefore, A ⊂ f- 1 ( f ( A )). (iii). For f ( x ) = sin x, A = { } , we have f- 1 ( f ( A )) = f- 1 (0) = { kπ : k = 0 , ± 1 , ± 2 , ···} 6 = A = { } . #2. Let f ( x ) be a continuous function on R 1 , and let B denote the Borel σ-algebra on R 1 . Show that M := f- 1 ( B ) := { E : E = f- 1 ( F ) for some F ∈ B} is a σ-algebra contained in B ....
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s3 - Math 8601 REAL ANALYSIS Fall 2010 Homework#3 Problems...

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