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| In the field of [[mathematics]] known as [[convex analysis]], the '''characteristic function''' of a set is a [[convex function]] that indicates the membership (or non-membership) of a given element in that set. It is similar to the usual [[indicator function]], and one can freely convert between the two, but the characteristic function as defined below is better-suited to the methods of convex analysis.
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| ==Definition==
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| Let <math>X</math> be a [[set (mathematics)|set]], and let <math>A</math> be a [[subset]] of <math>X</math>. The '''characteristic function''' of <math>A</math> is the function
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| :<math>\chi_{A} : X \to \mathbb{R} \cup \{ + \infty \}</math>
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| taking values in the [[extended real number line]] defined by
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| :<math>\chi_{A} (x) := \begin{cases} 0, & x \in A; \\ + \infty, & x \not \in A. \end{cases}</math>
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| ==Relationship with the indicator function==
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| Let <math>\mathbf{1}_{A} : X \to \mathbb{R}</math> denote the usual indicator function:
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| :<math>\mathbf{1}_{A} (x) := \begin{cases} 1, & x \in A; \\ 0, & x \not \in A. \end{cases}</math> | |
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| If one adopts the conventions that
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| * for any <math>a \in \mathbb{R} \cup \{ + \infty \}</math>, <math>a + (+ \infty) = + \infty</math> and <math>a (+\infty) = + \infty</math>;
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| * <math>\frac{1}{0} = + \infty</math>; and
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| * <math>\frac{1}{+ \infty} = 0</math>;
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| then the indicator and characteristic functions are related by the equations
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| :<math>\mathbf{1}_{A} (x) = \frac{1}{1 + \chi_{A} (x)}</math>
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| and
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| :<math>\chi_{A} (x) = (+ \infty) \left( 1 - \mathbf{1}_{A} (x) \right).</math>
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| ==Bibliography==
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| * {{cite book
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| | last = Rockafellar
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| | first = R. T.
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| | authorlink = R. Tyrrell Rockafellar
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| | title = Convex Analysis
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| | publisher = Princeton University Press
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| | location = Princeton, NJ
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| | year = 1997
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| | origyear = 1970
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| | isbn = 978-0-691-01586-6
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| }}
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| [[Category:Convex analysis]]
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The writer is recognized by the title of Numbers Wunder. Bookkeeping is what I do. For years he's been residing in North Dakota and his family loves it. To do aerobics is a factor that I'm completely addicted to.
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