Step function

In mathematics, a function on the reals R is a step function if it can be written as a finite linear combination of characteristic functions of semi-open intervals.

Let the following quantities be given:

• a sequence of coefficients

  ${\displaystyle \{\alpha _{0},\dots ,\alpha _{n}\}\subset \mathbb {R} ,\;n\in \mathbb {N} \setminus \{0\}}$

• a sequence of interval margins

  ${\displaystyle \{x_{1}<\dots <x_{n-1}\}\subset \mathbb {R} }$

• a set of functions ${\displaystyle p_{i}:\Re \rightarrow \Re ,\;\;i\in {\overline {0,n}}}$ such that:

  ${\displaystyle p_{0}(x)=\left\{{\begin{matrix}1,&\mathrm {if} \;x<x_{0}\\0,&\mathrm {if} \;x\geq x_{0}\end{matrix}}\right.}$

  ${\displaystyle \forall i\in {\overline {1,n-1}}\;\;p_{i}(x)=\left\{{\begin{matrix}1,&\mathrm {if} \;x_{i-1}\leq x<x_{i}\\0,&\mathrm {otherwise} \end{matrix}}\right.}$

  ${\displaystyle p_{n}(x)=\left\{{\begin{matrix}0,&\mathrm {if} \;x\leq x_{n-1}\\1,&\mathrm {if} \;x>x_{n-1}\end{matrix}}\right.}$

Definition: Given the notations above, a function f:R→R is a step function if and only if it could be written as

${\displaystyle f(x)=\sum \limits _{i=0}^{n}\alpha _{i}\cdot p_{i}(x).}$

Note: Step functions are constant on each interval ${\displaystyle [x_{i-1},x_{i}).}$

Special step functions

Heaviside step function

Related topics

• simple function
• piecewise function