singular function


A monotoneMathworldPlanetmath, non-constant, functionMathworldPlanetmath f:[a,b] is said to be a singular function (or a purely singular function) if f(x)=0 almost everywhere.

It is easy to see that a singular function cannot be absolutely continuousMathworldPlanetmath ( If an absolutely continuous function f:[a,b] satisfies f(x)=0 almost everywhere, then it must be constant.

An example of such a function is the famous Cantor functionMathworldPlanetmath. 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).


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


  • 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