Uniformly Continuous but not absolutely continuous example?

**Mimi89** · March 9th 2010, 05:33 AM

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.

**Opalg** · March 9th 2010, 01:23 PM

Originally Posted by Mimi89

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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