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 intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces.

Definition and first consequences

A function is called a step function if it can be written as where n\ge 0, \alpha_i are real numbers, A_i are intervals, and \chi_A is the indicator function of A: In this definition, the intervals A_i can be assumed to have the following two properties: Indeed, if that is not the case to start with, a different set of intervals can be picked for which these assumptions hold. For example, the step function can be written as

Variations in the definition

Sometimes, the intervals are required to be right-open or allowed to be singleton. The condition that the collection of intervals must be finite is often dropped, especially in school mathematics, though it must still be locally finite, resulting in the definition of piecewise constant functions.

Examples

sgn(x) , which is −1 for negative numbers and +1 for positive numbers, and is the simplest non-constant step function. H(x) , which is 0 for negative numbers and 1 for positive numbers, is equivalent to the sign function, up to a shift and scale of range. It is the mathematical concept behind some test signals, such as those used to determine the step response of a dynamical system.

Non-examples

Properties