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 such that

Absolutely continuous: A function F is absolutely continuous on [a,b] if given so that whenever at intervals all disjoint.

P.S. I also posted this in "Other Advanced Topics"; should I delete it and how? (I think this subforum is better suited for the question)