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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 Spring '11
 NA
 Topology, Continuous function, Metric space, Lebesgue measure, Lebesgue integration

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