UMD Analysis Qualifying Exam/Jan11 Real

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Problem 1[edit | edit source]

Let be an absolutely continuous function on [0,1] with . Prove that .

Solution[edit | edit source]

Since is (absolutely) continuous on [0,1] with then there exists some .

Since then for any there exists some such that for any finite collection of disjoint intervals such that if then .

Then for any such collection of intervals described above, we have .