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Ma28_Final_Exam

# Ma28_Final_Exam - X Problem 6 Let f a,b → R be continuous...

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Math 28 2009: Final Exam Instructions: Problem 1. Let ( a n ) R be a sequence. Let A be the set of limit points of subsequences of ( a n ). In other words, a A if and only if there exists a subsequence ( a n k ) such that ( a n k ) a . Show that A is closed. Problem 2. Determine the cardinality of the set { ( x, y ) | x, y Q , x 2 + y 2 = 1 } . (Hint: consider lines through (0 , 1).) Problem 3. Recall that we say f : R R is periodic with period T if f ( x + T ) = f ( x ) for all x . (a) Show that if f is continuous and periodic then it attains its supremum and infimum. (b) Prove that any function that is continuous and periodic must be uniformly continuous. Problem 4. Let A, B R be nonempty disjoint compact sets. Show that A B is not connected. Problem 5. Let ( X, d ) be a metric space and let f n : X X be uniformly continuous for all n N . Show that if ( f n ) converges uniformly on X , then the limit function is also uniformly continuous on
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Unformatted text preview: X . Problem 6. Let f : ( a,b ) → R be continuous. Prove that given x 1 ,...,x n in ( a,b ) that there exists an x ∈ ( a,b ) such that f ( x ) = 1 n ( f ( x 1 ) + ··· + f ( x n )) . Problem 7. Let f : ( a,b ) → R . Given c ∈ ( a,b ), show that f is diﬀerentiable at c if and only if there exists a constant M so that f ( x ) = f ( c ) + M ( x-c ) + r ( x ) where r ( x ) satisﬁes lim x → c r ( x ) x-c = 0 . Problem 8. Suppose that the series ∑ ∞ n =1 f n ( x ) converges uniformly on A and that g : A → R is bounded. (a) Prove that the series ∑ ∞ n =1 g ( x ) f n ( x ) converges uniformly on A . (b) Show by example that the boundedness of g is necessary for part (a). 1...
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