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A construct an example of a function f01 which is

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(a) Construct an example of a function f:[0,1] which is continuous nowhere and a function g:[0,1] which is continuous on all of [0,1], such that m({x: f(x) g(x)}) = 0. (b) Let <q n > be an enumeration of all the rational numbers in the interval [0,1]. Define f:[0,1] by the following formula: f(x) = 2 -n . q n x Here, of course, the sum is over the indices n such that q n is no larger than x. Note that since the geometric series 2 -n dominates the sum defining f, convergence is not a problem. Show f is continuous at each irrational number in [0,1] and discontinuous at each rational number in [0,1]. Can you find a continuous function g:[0,1] such that the measure of the set {x: f(x) g(x)} is zero? Proof?? Hint: (1) For (a) there are a couple of very easy and obvious functions that do the job. (2) For (b), it helps to observe that the discontinuities are jumps since f is increasing.
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MAA5616/FINAL EXAM/PART C Autumn, 1997 Page 6 of 6 6. (a) Let f n (x) = n -1 χ [-n,n] (x) for n ε and x ε . Show {f n } converges uniformly to f(x)=0o n . (b) With proof determine whether there is a Lebesgue integrable function g defined on such the g(x) f n (x) for every x ε and n ε .
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