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413finalexamdraft

# 413finalexamdraft - MATH 413 FINAL EXAM Math 413 nal exam...

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MATH 413 FINAL EXAM Math 413 final exam, 13 May 2008. The exam starts at 9:00 am and you have 150 minutes. No textbooks or calculators may be used during the exam. This exam is printed on both sides of the paper. Good luck! (1) (20 marks) Let X = (0 , 1] R . State whether each of the following statements about X is true or false, giving a brief reason for each answer. (a) X is bounded. (b) X can be written as a countable union of open sets. (c) X is compact. (d) There is a point x 0 X at which the function f ( x ) = log( x ) + x 5 - 8 x 4 - 3 achieves its supremum on X (that is, f ( x 0 ) = sup { f ( x ) : x X } ). (2) (20 marks) Let A R . Recall that a function f : A R is said to satisfy a Lipschitz condition on A if there is some M R such that | f ( x ) - f ( y ) | ≤ M | x - y | for all x, y A . (a) Let n N . Show that the function f n : [0 , 1] R defined by f n ( x ) = q x + 1 n satisfies a Lipschitz condition on [0 , 1]. (Hint: you may wish to use the fact that for all a, b > 0, ( a + b )( a - b ) = a - b .) (b) Show that the sequence of functions { f n } converges uniformly on [0 , 1] to the

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413finalexamdraft - MATH 413 FINAL EXAM Math 413 nal exam...

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