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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 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>

  (Although the intervals are shown as being closed below and open above, this is not necessary to the definition; all that is required is that the intervals A_n do not intersect, and that their union is the set of real numbers.)

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>.

[edit] Special step functions

A particular step function, the unit step function or Heaviside step function H(x), is obtained by setting n=1, α₀=0, α₁=1, and x₁=0 in the general expression above. It is the mathematical concept behind some test signals, as those used to determine the step response of a dynamical system.

[edit] See also

• simple function
• piecewise defined functionde:Treppenfunktion

ro:Funcţie scară sv:Stegfunktion tr:Basamak Fonksiyonu zh:阶跃函数