Step Function: Simple Definition, Examples

Calculus Definitions >

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:

Where:

• n is an integer greater than zero,
• α_i are real numbers.
• A_i are intervals,
• Χ_A is the indicator function, defined as:
  

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.


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

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.

References

Step and Box Functions. Retrieved from https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-iii-fourier-series-and-laplace-transform/step-and-delta-functions-integrals-and-generalized-derivatives/MIT18_03SCF11_s24_1text.pdf on November 25, 2018.
Labossiere, P. ME354 Mechanics of Materials Laboratory. Beam deflections using discontinuity functions. Retrieved OCtober 7, 2019 from: http://courses.washington.edu/me354a/discontinuity_functions.pdf

Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!