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Thread: Uniformly Continuous but not absolutely continuous example?

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    Uniformly Continuous but not absolutely continuous example?

    Could you please help me find an example of the following:

    a function which is uniformly continuous but not absolutely continuous.

    Definitions:

    Uniformly continuous: A function f is uniformly continuous if  \forall \epsilon > 0 \ \exists \delta > 0 such that  | x-x_0 | < \delta \rightarrow | f(x) - f(x_0) | < \epsilon

    Absolutely continuous: A function F is absolutely continuous on [a,b] if given  \epsilon >0 \ \exists \delta > 0 so that  \sum\limits_{k+1}^{N} { | F( b_k) - F(a_k) |} < \epsilon whenever  \sum\limits_{k=1}^{N} {b_k - a_k} < \delta at intervals  (a_k, b_k) all disjoint.
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  2. #2
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    Quote Originally Posted by Mimi89 View Post
    Could you please help me find an example of the following:

    a function which is uniformly continuous but not absolutely continuous.
    As a quick guess, the Cantor staircase ought to have those properties.
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