Singular Function

See: Singular, Function, Absolute Continuity.

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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/).
  • 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).

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