UMD Analysis Qualifying Exam/Jan11 Real

Problem 1 edit

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

Solution edit

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  .