In mathematics, a function on the real numbers is called a step function (or staircase 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
- for all real numbers
where are real numbers, are intervals, and is the indicator function of :
In this definition, the intervals can be assumed to have the following two properties:
- The intervals are disjoint, for
- The union of the intervals is the entire real line,
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
- A constant function is a trivial example of a step function. Then there is only one interval,
- The Heaviside function H(x) is an important step function. It is the mathematical concept behind some test signals, such as those used to determine the step response of a dynamical system.
- The rectangular function, the normalized boxcar function, is the next simplest step function, and is used to model a unit pulse.
- The integer part function is not a step function according to the definition of this article, since it has an infinite number of "steps".
- The sum and product of two step functions is again a step function. The product of a step function with a number is also a step function. As such, the step functions form an algebra over the real numbers.
- A step function takes only a finite number of values. If the intervals in the above definition of the step function are disjoint and their union is the real line, then for all
- The Lebesgue integral of a step function is where is the length of the interval and it is assumed here that all intervals have finite length. In fact, this equality (viewed as a definition) can be the first step in constructing the Lebesgue integral.
- The derivative of a step function is the Dirac delta function