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Math Help - dirichlets function

  1. #1
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    dirichlets function

    prove that dirichlets function (nowhere continuous function on [0,1]) is not of bounded variation.
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    If a function is of bounded variation then it is the difference of two monotone functions and is therefore continuous almost everywhere. You will find a bit more detail about those assertions and their proofs here.
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    dirichlets function

    how can i show that dirichlets function can not be expressed as the diffrence of two increasing functions?
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    Quote Originally Posted by Kat-M View Post
    how can i show that dirichlets function can not be expressed as the diffrence of two increasing functions?
    That's because the difference of two increasing functions would have to be continuous except at countably many points (see Monotonic function - Wikipedia, the free encyclopedia).
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