# Singular Function

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**See:** Singular, Function, Absolute Continuity.

## References

- http://en.wikipedia.org/wiki/Singular_function
- In mathematics, a
**singular function**is any function ƒ(*x*) defined on the interval [*a*,*b*] that has the following properties:- ƒ(
*x*) is continuous on [*a*,*b*]. - there exists a set
*N*of measure 0 such that for all*x*outside of*N*the derivative ƒ ′(*x*) exists and is zero, that is, the derivative of [math]\displaystyle{ f }[/math] vanishes almost everywhere. - ƒ(
*x*) is nondecreasing on [*a*,*b*]. - ƒ(
*a*) <*ƒ*(*b/).*

- ƒ(
- A standard example of a singular function is the Cantor function, which is sometimes called the
**devil's staircase**(a term also used for singular functions in general). There are, however, other functions that have been given that name. One is defined in terms of the circle map. - If ƒ(
*x*) = 0 for all [math]\displaystyle{ x }[/math] =*a*and ƒ(*x*) = 1 for all*x*=*b*, then the function can be taken to represent a cumulative distribution function for a random variable which is neither a discrete random variable (since the probability is zero for each point) nor an absolutely continuous random variable (since the probability density is zero everywhere it exists).

- In mathematics, a