RealAnalysis2010jan - absolute continuity for f on I(b...

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Real analysis qualifying exam Jan. 13, 2010 1. (a) Let f be a continuous map of a metric space X into a metric space Y . True or False. If false either give a counterexample, or make the statement true by either adding a hypothesis or modifying the conclusion. Do not prove if true. (i) If X is compact, then so is f ( X ) . (ii) If X is connected, then so is f ( X ) . (iii) If f is one-to-one, then f 1 : f ( X ) X is continuous. (b) The Cantor set C [0 , 1] R consists of all sums x = j =1 n j 3 j where the n j are allowed to form any sequence of 0 ’s and 2 ’s. Let f : C [0 , 1] be the canonical map defined by f ( x ) = 1 2 j =1 n j 2 j . Prove or Disprove. (i) f is onto. (ii) f is continuous. (iii) f is one-to-one. 2. Let { f j } be a sequence of Lebesgue measurable functions that converges pointwise a.e. to a function f on the interval I = [0 , 1] . Let F L p ( I ) and g L p ( I ) where p and p are dual exponents, 1 p ≤ ∞ . (a) If p > 1 , f j p 1 ( j = 1 , 2 , . . . ) and I f j g I Fg , prove that I fg = I Fg . (b) Show by example that the conclusion of part (a) is false when p = 1 . 3. Let f be a real valued function on the interval I = [ a, b ] . (a) Give the definition of
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Unformatted text preview: absolute continuity for f on I . (b) Suppose f is absolutely continuous on I . True or False. If false either give a counterexample or modify the statement so that it is true. Do not prove if true. (i) f is uniformly continuous on I . (ii) f is differentiable at every x in the interior of I . (iii) f ′ ∈ L 1 ( I ) and f ( x ) − f ( a ) = ∫ x a f ′ ( t ) dt , a ≤ x ≤ b . (c) Suppose f is absolutely continuous on I . Prove that the set of values { y = f ( x ) : f ′ ( x ) is not defined } has measure zero . (d) Suppose f is absolutely continuous on I . Prove that the set of values { y = f ( x ) : f ′ ( x ) = 0 } has measure zero . 4. Let Borel functions f ∈ L 1 ( R ) and g ∈ L 1 ( R ) be given so that f ( x − y ) g ( y ) is a Borel function on R 2 . Prove that ∫ ∞ −∞ | f ( x − y ) g ( y ) | dy < ∞ for a.e. x ....
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