Step Function

In Mathematics, a step function (also called as staircase function) is defined as a piecewise constant function, that has only a finite number of pieces. In other words, if a function on the real numbers can be described as a finite linear combination of indicator functions of given intervals. It is also called a floor function or greatest integer function. Also, it is a discontinuous function.

Definition:

A step function f: R → R can be written in the form:

Step function formula 1

for all real numbers x.

If n ≥ 0, αi are real numbers and Ai are intervals, then the indicator function of A is χA, and it can be written as below:

Step function formula 2

(Or)

A function f: R → R is called a step or greatest integer function if y = f(x) = [x] for x ∈ R.

Read more:

Definite Integral

Operations of Integers

Real Functions

Relations and Functions Class 11

Domain and Range

In a step function for each value of x, f(x) takes the value of the greatest integer, less than or equal to x. For example:

[-2.19] = -3

[3.67] = 3

[-0.83] = -1

The domain of this function is a group of real numbers that are divided into intervals such as [-5, 3), [-4, 2), [-3, 1), [-2, 0) and so on. This explains the domain and range relations of a step function.

This can be generalized as below:

[x] = -2, -2 ≤ x < -1

[x] = -1, -1 ≤ x < 0

[x] = 0, 0 ≤ x < 1

[x] = 1, 1 ≤ x < 2

Step Function Graph

The graph of a step function is as shown below.

Step function graph

Step Function Examples

Example 1:

Draw a graph of the step function:

Example function 1

Solution:

-2, 0, 2 are the values of y.

x < -1 means the values of x = …, -4, -3, -2, -1

-1 ≤ x ≤ 2 means, x = -1, 0, 1, 2

x > 1 means the values of x = 1, 2, 3, 4, …..

Step function graph 2

The above graph is viewed as a group of steps and hence it is also called a step function graph. The left endpoint in every step is blocked(dark dot) to show that the point is a member of the graph, and the other right end arrow indicates that the values are infinite. That means only definite values are shown with dark dots.

Properties

The important properties of step functions are given below:

  • The sum or product of two-step functions is also a step function.
  • If a step function is multiplied by a number, then the result produced is again a step function. That indicates the step functions create an algebra over the real numbers
  • A step function can take only a finite number of values
  • Piecewise linear function is the definite integral of a step function

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