# mt1a - Math 8601 REAL ANALYSIS Fall 2010 Some problems for...

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Unformatted text preview: Math 8601: REAL ANALYSIS. Fall 2010 Some problems for Midterm Exam #1 on Wednesday, October 6. You will have 50 minutes (10:10 am–11:00 am) to work on 5 problems, 2 of which will be selected from the following list. It is recommended to prepare solutions of take-home problems on separate pages, then you can just enclose them together with in-class problems without rewriting. No books, no notes during this exam. Calculators are permitted, however, for full credit, you need to show step-by-step calculations. #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 . Hint. You can use without proof the properties of f- 1 on p.4, and Proposition 0.23 on p.14....
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