singular function

Definition.

A monotone, non-constant, function $f\colon[a,b]\to{\mathbb{R}}$ is said to be a singular function (or a purely singular function) if $f^{\prime}(x)=0$ almost everywhere.

It is easy to see that a singular function cannot be absolutely continuous (http://planetmath.org/AbsolutelyContinuousFunction2): If an absolutely continuous function $f\colon[a,b]\to\mathbb{R}$ satisfies $f^{\prime}(x)=0$ almost everywhere, then it must be constant.

An example of such a function is the famous Cantor function. While this is not a strictly increasing function, there also do exist singular functions which are in fact strictly increasing, and even more amazingly functions that are quasisymmetric (see attached example).

Theorem.

Any monotone increasing function can be written as a sum of an absolutely continuous function and a singular function.

References

• 1 H. L. Royden. . Prentice-Hall, Englewood Cliffs, New Jersey, 1988
 Title singular function Canonical name SingularFunction Date of creation 2013-03-22 14:08:05 Last modified on 2013-03-22 14:08:05 Owner jirka (4157) Last modified by jirka (4157) Numerical id 11 Author jirka (4157) Entry type Definition Classification msc 26A30 Synonym purely singular function Related topic AbsolutelyContinuousFunction2 Related topic CantorFunction Related topic CantorSet Related topic AbsolutelyContinuousFunction2 Defines singular function