examples of characteristic functions
In this entry, we give some examples of characteristic functions written in closed form (i.e. (http://planetmath.org/Ie), not piecewise defined).
We use the following conventions:

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$X$ denotes the domain (http://planetmath.org/Domain) of the characteristic function^{}

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$A$ denotes a subset of $X$

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${\chi}_{A}$ denotes the characteristic function of $A$

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$\cdot $ denotes distance from $0$ (e.g. (http://planetmath.org/Eg) absolute value^{} or modulus^{} (http://planetmath.org/Modulus2))

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$\lfloor \cdot \rfloor $ denotes the floor function

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$\lceil \cdot \rceil $ denotes the ceiling function

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$\mathrm{Re}$ denotes the real part^{} of a complex number^{}

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$\mathrm{Im}$ denotes the imaginary part of a complex number

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$\mathbb{N}$ denotes the positive integers

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$\mu $ denotes the Möbius function^{}

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$\phi $ denotes the Euler totient function

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${\mathcal{O}}_{K}$ denotes the ring of integers of a number field^{} $K$

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${\mathrm{N}}_{K}$ denotes the norm on a number field $K$
Below is a of characteristic functions:
Title  examples of characteristic functions 

Canonical name  ExamplesOfCharacteristicFunctions 
Date of creation  20130322 17:54:28 
Last modified on  20130322 17:54:28 
Owner  Wkbj79 (1863) 
Last modified by  Wkbj79 (1863) 
Numerical id  11 
Author  Wkbj79 (1863) 
Entry type  Example 
Classification  msc 11R04 
Classification  msc 11A25 
Classification  msc 26A09 
Classification  msc 2600 
\@unrecurse 