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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.

Bibliography

1
H. L. Royden. Real Analysis. Prentice-Hall, Englewood Cliffs, New Jersey, 1988



"singular function" is owned by jirka.
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See Also: absolutely continuous function, Cantor function, Cantor set, absolutely continuous function

Other names:  purely singular function
Also defines:  singular function

Attachments:
example of a strictly increasing quasisymmetric singular function (Example) by jirka
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Cross-references: sum, monotone increasing, quasisymmetric, even, strictly increasing, Cantor function, absolutely continuous function, easy to see, almost everywhere, function, monotone
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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 7934 times total.

Classification:
AMS MSC26A30 (Real functions :: Functions of one variable :: Singular functions, Cantor functions, functions with other special properties)

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