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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


Where:

  • 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

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

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Stephanie Glen. "Step Function: Simple Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/step-function/
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