Step Function

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 intervals
: $A_0 := (-\infty, x_1)$
: $A_i := [x_i, x_{i+1})$ (for $i=1,\cdots,n-2$ )
: $A_n := [x_{n-1},\infty)$

Definition: Given the notations above, a function

f: \mathbb{R} 
ightarrow \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

A

:
:

1_A(x) =
\left\{
  \begin{matrix}
    1, & \mathrm{if} \; x \in A \ 
    0, & \mathrm{otherwise} 
  \end{matrix}

ight.

Note: for all

i=0,\cdots,n

and

x \in A_i

it holds:

f(x)=\alpha_i

SPECIAL STEP FUNCTIONS

Heaviside Step Function

SEE ALSO

	CATEGORIES ABOUT STEP FUNCTION

	elementary special functions

Information About