WOLFRAM

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)Summary of the most common use cases

Evaluate numerically:

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Plot in one dimension:

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Plot in two dimensions:

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UnitStep is a piecewise function:

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Scope  (34)Survey of the scope of standard use cases

Numerical Evaluation  (6)

Evaluate numerically:

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UnitStep always returns an exact result:

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Evaluate efficiently at high precision:

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UnitStep can deal with realvalued intervals:

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Compute the elementwise values of an array using automatic threading:

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Or compute the matrix UnitStep function using MatrixFunction:

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Compute average-case statistical intervals using Around:

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Specific Values  (4)

Value at zero:

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Value at infinity:

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Evaluate for symbolic parameters:

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

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Visualization  (4)

Plot the UnitStep function:

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Visualize shifted UnitStep functions:

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Visualize the composition of UnitStep with a periodic function:

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Plot UnitStep in three dimensions:

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Function Properties  (10)

Function domain of UnitStep:

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It is restricted to real inputs:

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Function range of UnitStep:

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UnitStep has a jump discontinuity at the point :

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UnitStep is not an analytic function:

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It has both singularities and discontinuities:

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UnitStep is nondecreasing:

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UnitStep is not injective:

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UnitStep is not surjective:

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UnitStep is non-negative:

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UnitStep is neither convex nor concave:

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TraditionalForm formatting:

Differentiation and Integration  (6)

First derivative with respect to x:

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All higher-order derivatives the same:

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First derivative with respect to z:

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Compute the indefinite integral using Integrate:

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Verify the anti-derivative away from the singular point:

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Definite integral:

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Integral over an infinite domain:

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Integral Transforms  (4)

FourierTransform of UnitStep:

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FourierSeries:

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Find the LaplaceTransform of UnitStep:

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The convolution of UnitStep with itself:

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Applications  (8)Sample problems that can be solved with this function

Generate a square wave:

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Compute a step response for a continuous-time system:

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Using transform methods:

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Compute a step response for a discrete-time system:

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Using transform methods:

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Solve the timeindependent Schrödinger equation with piecewise analytic potential:

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

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Here is the resulting probability plotted:

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Construct the Walsh function:

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Define a BoseEinstein and a MaxwellBoltzmann distribution function with UnitStep and Exp:

Plot the distributions:

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

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Properties & Relations  (4)Properties of the function, and connections to other functions

The derivative of UnitStep is a piecewise function:

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The derivative of HeavisideTheta is a distribution:

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Expand into UnitStep of linear factors:

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Convert into Piecewise:

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Integrate over finite and infinite domains:

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Possible Issues  (3)Common pitfalls and unexpected behavior

Symbolic preprocessing of functions containing UnitStep can be timeconsuming:

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

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Differentiating Abs does not yield UnitStep:

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

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

Out[3]=3
Wolfram Research (1999), UnitStep, Wolfram Language function, https://reference.wolfram.com/language/ref/UnitStep.html (updated 2007).
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).

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.

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

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: 19-June-2025 ]}

@misc{reference.wolfram_2025_unitstep, author="Wolfram Research", title="{UnitStep}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/UnitStep.html}", note=[Accessed: 19-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_unitstep, organization={Wolfram Research}, title={UnitStep}, year={2007}, url={https://reference.wolfram.com/language/ref/UnitStep.html}, note=[Accessed: 19-June-2025 ]}

@online{reference.wolfram_2025_unitstep, organization={Wolfram Research}, title={UnitStep}, year={2007}, url={https://reference.wolfram.com/language/ref/UnitStep.html}, note=[Accessed: 19-June-2025 ]}