singular function

Definition.

A monotone, non-constant, function $f\\colon[a,b]\\to{\\mathbb{R}}$ is said to be asingular function (or a purely singular function) if $f^{\\prime}(x)=0$ almost everywhere.

It is easy to see that a singular function cannot beabsolutely continuous (http://planetmath.org/AbsolutelyContinuousFunction2):If an absolutely continuous function $f\\colon[a,b]\\to\\mathbb{R}$ 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

单词	SingularFunction
释义	singular function Definition. A monotone, non-constant, function $f\\colon[a,b]\\to{\\mathbb{R}}$ is said to be asingular function (or a purely singular function) if $f^{\\prime}(x)=0$ almost everywhere. It is easy to see that a singular function cannot beabsolutely continuous (http://planetmath.org/AbsolutelyContinuousFunction2):If an absolutely continuous function $f\\colon[a,b]\\to\\mathbb{R}$ 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
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