# Thread: Uniformly Continuous but not absolutely continuous example?

1. ## 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 $\displaystyle \forall \epsilon > 0 \ \exists \delta > 0$ such that $\displaystyle | x-x_0 | < \delta \rightarrow | f(x) - f(x_0) | < \epsilon$

Absolutely continuous: A function F is absolutely continuous on [a,b] if given $\displaystyle \epsilon >0 \ \exists \delta > 0$ so that $\displaystyle \sum\limits_{k+1}^{N} { | F( b_k) - F(a_k) |} < \epsilon$ whenever $\displaystyle \sum\limits_{k=1}^{N} {b_k - a_k} < \delta$ at intervals $\displaystyle (a_k, b_k)$ all disjoint.

2. 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.