singular function

Definition.

A monotone, non-constant, function $f:[a,b]\to ℝ$ 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 $f:[a,b]\to ℝ$ 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.

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