|
|
|
|
singular function
|
(Definition)
|
|
Definition 1 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 $f'(x) = 0$ almost everywhere.
It is easy to see that a singular function cannot be absolutely continuous: If an absolutely continuous function $f \colon [a,b] \to \mathbb{R}$ satisfies $f'(x)=0$ 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 1 Any monotone increasing function can be written as a sum of an absolutely continuous function and a singular function.
- 1
- H. L. Royden. Real Analysis. Prentice-Hall, Englewood Cliffs, New Jersey, 1988
|
"singular function" is owned by jirka.
|
|
(view preamble | get metadata)
Cross-references: sum, monotone increasing, quasisymmetric, even, strictly increasing, Cantor function, absolutely continuous function, easy to see, almost everywhere, function, monotone
There are 2 references to this entry.
This is version 8 of singular function, born on 2004-02-05, modified 2006-09-17.
Object id is 5548, canonical name is SingularFunction.
Accessed 9348 times total.
Classification:
| AMS MSC: | 26A30 (Real functions :: Functions of one variable :: Singular functions, Cantor functions, functions with other special properties) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
