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 takehome problems on separate pages, then you can just enclose them together with inclass problems without rewriting. No books, no notes during this exam. Calculators are permitted, however, for full credit, you need to show stepbystep 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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