In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces.
Definition and first consequences A function f : R → R {\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} } is called a step function if it can be written as
f ( x ) = ∑ i = 0 n α i χ A i ( x ) {\displaystyle f(x)=\sum \limits _{i=0}^{n}\alpha _{i}\chi _{A_{i}}(x)} , for all real numbers x {\displaystyle x}
where n ≥ 0 {\displaystyle n\geq 0} , α i {\displaystyle \alpha _{i}} are real numbers, A i {\displaystyle A_{i}} are intervals, and χ A i {\displaystyle \chi _{A_{i}}} is the indicator function of A i {\displaystyle A_{i}} :
χ A i ( x ) = { 1 if x ∈ A i 0 if x ∉ A i {\displaystyle \chi _{A_{i}}(x)={\begin{cases}1&{\text{if }}x\in A_{i}\\0&{\text{if }}x\notin A_{i}\\\end{cases}}}
In this definition, the intervals A i {\displaystyle A_{i}} can be assumed to have the following two properties:
The intervals are pairwise disjoint: A i ∩ A j = ∅ {\displaystyle A_{i}\cap A_{j}=\emptyset } for i ≠ j {\displaystyle i\neq j}
The union of the intervals is the entire real line: ⋃ i = 0 n A i = R . {\displaystyle \bigcup _{i=0}^{n}A_{i}=\mathbb {R} .}
Indeed, if that is not the case to start with, a different set of intervals can be picked for which these assumptions hold. For example, the step function
f = 4 χ [ − 5 , 1 ) + 3 χ ( 0 , 6 ) {\displaystyle f=4\chi _{[-5,1)}+3\chi _{(0,6)}}
can be written as
f = 0 χ ( − ∞ , − 5 ) + 4 χ [ − 5 , 0 ] + 7 χ ( 0 , 1 ) + 3 χ [ 1 , 6 ) + 0 χ [ 6 , ∞ ) . {\displaystyle f=0\chi _{(-\infty ,-5)}+4\chi _{[-5,0]}+7\chi _{(0,1)}+3\chi _{[1,6)}+0\chi _{[6,\infty )}.}
Variations in the definition Sometimes, the intervals are required to be right-open or allowed to be singleton. The condition that the collection of intervals must be finite is often dropped, especially in school mathematics, though it must still be locally finite, resulting in the definition of piecewise constant functions.
Examples
A constant function is a trivial example of a step function. Then there is only one interval, A 0 = R . {\displaystyle A_{0}=\mathbb {R} .}
… excerpt ends here. Continue reading the full article.
