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331fin

# 331fin - B U Department of Mathematics Fall 2005 Math 331...

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B U Department of Mathematics Fall 2005 Math 331 - Real Analysis I, Final Exam, 5/1/2006, 15:00-17:30 Whatever theorem, proposition, lemma, exercise, example you use, you must EXACTLY STATE it Full Name : (its hypothesis and its conclusion). Besides, either it MUST have been already proven in the class or Student ID : you must PROVE it now. If you do not state what you are using or do not prove, your solutions will not be accepted. I encourage you make use of theorems. Do not forget to return this exam sheet! over 60 Below, C ( X, Y ) and C b ( X, Y ) denote the set of all continuous functions (and bounded in the latter) from X to Y equipped with the sup metric whenever it makes sense. The answers to questions 1-3 are to be written on this sheet just below the question. 1. (1) What do we mean when we say ”pointwise” and ”uniform”? 2. (2) Sketch a closed subset C of R 2 such that π ( C ) R is not closed, where π : R 2 R is the projection given by π ( x, y ) = x (which is a continuous function). 3. (3) Let

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331fin - B U Department of Mathematics Fall 2005 Math 331...

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