Step function

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
$\{\alpha_0, \dots, \alpha_n\}\subset \mathbb{R},\; n \in \mathbb{N} \setminus \{0\}$
a sequence of interval margins
$\{x_1 < \dots < x_{n-1}\} \subset \mathbb{R}$
a sequence of intervals
$A_0 := (-\infty, x_1)$
$A_i := [x_i, x_{i+1})$ (for $i=1,\cdots,n-2$ )
$A_n := [x_{n-1},\infty)$
(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 $f: \mathbb{R} \rightarrow \mathbb{R}$ is a step function if and only if it can be written as

f(x) = \sum\limits_{i=0}^n \alpha_i \cdot 1_{A_i}(x) for all

x \in \mathbb{R}

where

1_A

is the indicator function of

A

1_A(x) =

\left\{ \begin{matrix} 1, & \mathrm{if} \; x \in A \\ 0, & \mathrm{otherwise}. \end{matrix} \right.

Note: for all $i=0,\cdots,n$ and $x \in A_i$ it holds: $f(x)=\alpha_i.$

Special step functions

A version of the unit step function or Heaviside step function, H₁(x), is the special case n=1, α₀=0, α₁=1, and x₁=0.

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Special step functions

See also