Simple function
In the mathematical field of real analysis, a simple function is a real (or complex)-valued function over a subset of the real line, similar to a step function. Simple functions are sufficiently "nice" that using them makes mathematical reasoning, theory, and proof easier. For example, simple functions attain only a finite number of values. Some authors also require simple functions to be measurable; as used in practice, they invariably are.
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- enIn the mathematical field of real analysis, a simple function is a real (or complex)-valued function over a subset of the real line, similar to a step function. Simple functions are sufficiently "nice" that using them makes mathematical reasoning, theory, and proof easier. For example, simple functions attain only a finite number of values. Some authors also require simple functions to be measurable; as used in practice, they invariably are.
- Has abstract
- enIn the mathematical field of real analysis, a simple function is a real (or complex)-valued function over a subset of the real line, similar to a step function. Simple functions are sufficiently "nice" that using them makes mathematical reasoning, theory, and proof easier. For example, simple functions attain only a finite number of values. Some authors also require simple functions to be measurable; as used in practice, they invariably are. A basic example of a simple function is the floor function over the half-open interval [1, 9), whose only values are {1, 2, 3, 4, 5, 6, 7, 8}. A more advanced example is the Dirichlet function over the real line, which takes the value 1 if x is rational and 0 otherwise. (Thus the "simple" of "simple function" has a technical meaning somewhat at odds with common language.) All step functions are simple. Simple functions are used as a first stage in the development of theories of integration, such as the Lebesgue integral, because it is easy to define integration for a simple function and also it is straightforward to approximate more general functions by sequences of simple functions.
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- Simple function
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- enSimple function
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- Algebra over a field
- Borel σ-algebra
- Category:Measure theory
- Category:Real analysis
- Category:Types of functions
- Complex number
- Floor function
- Indicator function
- Integral
- Lebesgue integral
- Linear combination
- Mathematics
- Measurable function
- Measurable set
- Measure (mathematics)
- Pointwise
- Real analysis
- Real line
- Real number
- Sequence
- Serge Lang
- Sigma-algebra
- Step function
- Walter Rudin
- SameAs
- 545Mb
- Einfache Funktion
- Enkel funktion
- Fonction étagée
- Função simples
- Funkcja prosta
- Funzione semplice
- m.0642mv
- Q913022
- Simple function
- Проста функція
- Простая функция
- פונקציה פשוטה
- 単関数
- 简单函数
- 단순 함수
- Subject
- Category:Measure theory
- Category:Real analysis
- Category:Types of functions
- WasDerivedFrom
- Simple function?oldid=1119508798&ns=0
- WikiPageLength
- 4835
- Wikipage page ID
- 1890594
- Wikipage revision ID
- 1119508798
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- Template:Aut
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