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Step Function: Simple Definition, Examples

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step function example

Example of a step function (the red graph). This particular step function is right-continuous.

A step function features constant or “flat” areas punctuated by abrupt jumps, called jump discontinuities. It looks exactly like what the name suggests: it’s a series of one or more steps. More technically, it’s defined as a piecewise function, with a finite number of pieces.

Formal Definition

Formally, it is defined as a function over the real numbers that can be written as a finite linear combination of indicator functions of intervals.

It can also be defined as a function over the real numbers that can be written as:

function over the real numbers


  • n is an integer greater than zero,
  • αi are real numbers.
  • Ai are intervals,
  • ΧA is the indicator function, defined as:
    indicator function

Simple Examples of the Step Function

By the above definition, the constant function f(x) = A is a step function, though there is only one interval. In terms of “steps”, there’s just one; Imagine you’re standing on the first step of a set of stairs. There’s no up or down—just the flat surface you’re standing on.
Simple Examples of the Step Function

Another very simple step function—with a step in it—is the sign function. This function sends all positive numbers to +1, and all negative numbers to -1, so it has just one jump discontinuity.

the sign function

The sign function.

The rectangular function is constant at 0 for most values of x, but sends one interval Ai to a constant a.

example of step function

A rectangular function.

The Heaviside function is very like the sign function in that it has just discontinuity, at zero, but it sends all positive numbers to 1 and all negative numbers to zero.

heaviside function


Step and Box Functions. Retrieved from on November 25, 2018.
Labossiere, P. ME354 Mechanics of Materials Laboratory. Beam deflections using discontinuity functions. Retrieved OCtober 7, 2019 from:

Stephanie Glen. "Step Function: Simple Definition, Examples" From Calculus for the rest of us!

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