Step function
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In mathematics, a function on the real numbers is called step function if it can be written as a finite linear combination of indicator functions of half-open intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces.
Let the following quantities be given:
- a sequence of coefficients
- <math>\{\alpha_0, \dots, \alpha_n\}\subset \mathbb{R},\; n \in \mathbb{N} \setminus \{0\}</math>
- a sequence of interval margins
- <math>\{x_1 < \dots < x_{n-1}\} \subset \mathbb{R}</math>
- a sequence of intervals
- <math>A_0 := (-\infty, x_1)</math>
- <math>A_i := [x_i, x_{i+1})</math> (for <math>i=1,\cdots,n-2</math>)
- <math>A_n := [x_{n-1},\infty)</math>
Definition: Given the notations above, a function <math>f: \mathbb{R} \rightarrow \mathbb{R}</math> is a step function if and only if it can be written as
- <math>
f(x) = \sum\limits_{i=0}^n \alpha_i \cdot 1_{A_i}(x) </math> for all <math>x \in \mathbb{R}</math>. where <math>1_A</math> is the indicator function of <math>A</math>:
- <math>1_A(x) =
\left\{
\begin{matrix}
1, & \mathrm{if} \; x \in A \\
0, & \mathrm{otherwise}
\end{matrix}
\right. </math>
Note: for all <math>i=0,\cdots,n</math> and <math>x \in A_i</math> it holds: <math>f(x)=\alpha_i</math>