Step function

Encyclopedia : S : ST : STE : Step function

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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

  [\\subset \mathbb,\; n \in \mathbb \setminus \]

• a sequence of interval margins

  [\\} \subset \mathbb]

• a sequence of intervals

  [A_0 := (-\infty, x_1)]

  [A_i := [x_i, x_)] (for [i=1,\cdots,n-2])

  [A_n := [x_,\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 \rightarrow \mathbb] is a step function if and only if it can be written as

[f(x) = \sum\limits_^n \alpha_i \cdot 1_(x)] for all [x \in \mathbb] where [1_A] is the indicator function of [A]:

[1_A(x) =\left\ 1, & \mathrm \; x \in A \\ 0, & \mathrm. \end\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.

See also

• simple function
• piecewise function

 

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