# singular function

###### Definition.

A monotone^{}, non-constant, function^{} $f:[a,b]\to \mathbb{R}$ 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 $f:[a,b]\to \mathbb{R}$ 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 |