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step function: Definition and Much More from Answers.com

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.

Example of a step function with n=4.

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Example of a step function with n=4.

Let the following quantities be given:

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

Failed to parse (unknown function\limits): 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 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.

See also

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