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UnitStep: Unit step function—Wolfram Documentation

Differentiation and Integration  

UnitStep[x]

represents the unit step function, equal to 0 for and 1 for .

UnitStep[x1,x2,…]

represents the multidimensional unit step function which is 1 only if none of the are negative.

Details

  • Some transformations are done automatically when UnitStep appears in a product of terms.
  • UnitStep provides a convenient way to represent piecewise continuous functions.
  • UnitStep has attribute Orderless.
  • For exact numeric quantities, UnitStep internally uses numerical approximations to establish its result. This process can be affected by the setting of the global variable $MaxExtraPrecision.
  • UnitStep[] is 1.
  • UnitStep automatically threads over lists. »

Examples

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Basic Examples  (4)

Evaluate numerically:

Plot in one dimension:

Plot in two dimensions:

UnitStep is a piecewise function:

Scope  (34)

Numerical Evaluation  (6)

Evaluate numerically:

UnitStep always returns an exact result:

Evaluate efficiently at high precision:

UnitStep can deal with real‐valued intervals:

Compute the elementwise values of an array using automatic threading:

Or compute the matrix UnitStep function using MatrixFunction:

Compute average-case statistical intervals using Around:

Specific Values  (4)

Value at zero:

Value at infinity:

Evaluate for symbolic parameters:

Find a value of x for which the UnitStep[x]=1:

Visualization  (4)

Plot the UnitStep function:

Visualize shifted UnitStep functions:

Visualize the composition of UnitStep with a periodic function:

Plot UnitStep in three dimensions:

Function Properties  (10)

Differentiation and Integration  (6)

First derivative with respect to x:

All higher-order derivatives the same:

First derivative with respect to z:

Compute the indefinite integral using Integrate:

Verify the anti-derivative away from the singular point:

Definite integral:

Integral over an infinite domain:

Integral Transforms  (4)

Applications  (8)

Generate a square wave:

Compute a step response for a continuous-time system:

Using transform methods:

Compute a step response for a discrete-time system:

Using transform methods:

Solve the time‐independent Schrödinger equation with piecewise analytic potential:

This gives the probability of the random variable being in the interval :

Here is the resulting probability plotted:

Construct the Walsh function:

Define a Bose–Einstein and a Maxwell–Boltzmann distribution function with UnitStep and Exp:

Plot the distributions:

Find the representation of a mathematical expression with UnitStep in terms of FoxH:

Properties & Relations  (4)

The derivative of UnitStep is a piecewise function:

The derivative of HeavisideTheta is a distribution:

Expand into UnitStep of linear factors:

Convert into Piecewise:

Integrate over finite and infinite domains:

Possible Issues  (3)

Symbolic preprocessing of functions containing UnitStep can be time‐consuming:

Limit does not give UnitStep as a limit of smooth functions:

Differentiating Abs does not yield UnitStep:

Use RealAbs to get a derivative of absolute value on the reals:

But for the origin, where the derivative does not exist, this is equivalent to an expression in UnitStep:

Wolfram Research (1999), UnitStep, Wolfram Language function, https://reference.wolfram.com/language/ref/UnitStep.html (updated 2007).

Text

Wolfram Research (1999), UnitStep, Wolfram Language function, https://reference.wolfram.com/language/ref/UnitStep.html (updated 2007).

CMS

Wolfram Language. 1999. "UnitStep." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007. https://reference.wolfram.com/language/ref/UnitStep.html.

APA

Wolfram Language. (1999). UnitStep. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/UnitStep.html

BibTeX

@misc{reference.wolfram_2025_unitstep, author="Wolfram Research", title="{UnitStep}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/UnitStep.html}", note=[Accessed: 23-February-2026]}

BibLaTeX

@online{reference.wolfram_2025_unitstep, organization={Wolfram Research}, title={UnitStep}, year={2007}, url={https://reference.wolfram.com/language/ref/UnitStep.html}, note=[Accessed: 23-February-2026]}