UnitStep
✖
UnitStep
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
open allclose allBasic Examples (4)Summary of the most common use cases

https://wolfram.com/xid/0tz8czaa-boo3w3


https://wolfram.com/xid/0tz8czaa-c8v4qe


https://wolfram.com/xid/0tz8czaa-b2gxt5

UnitStep is a piecewise function:

https://wolfram.com/xid/0tz8czaa-cn25im

Scope (34)Survey of the scope of standard use cases
Numerical Evaluation (6)

https://wolfram.com/xid/0tz8czaa-l274ju


https://wolfram.com/xid/0tz8czaa-cksbl4


https://wolfram.com/xid/0tz8czaa-sj332k

UnitStep always returns an exact result:

https://wolfram.com/xid/0tz8czaa-hhwwys

Evaluate efficiently at high precision:

https://wolfram.com/xid/0tz8czaa-di5gcr


https://wolfram.com/xid/0tz8czaa-bq2c6r

UnitStep can deal with real‐valued intervals:

https://wolfram.com/xid/0tz8czaa-cslp3

Compute the elementwise values of an array using automatic threading:

https://wolfram.com/xid/0tz8czaa-thgd2

Or compute the matrix UnitStep function using MatrixFunction:

https://wolfram.com/xid/0tz8czaa-o5jpo

Compute average-case statistical intervals using Around:

https://wolfram.com/xid/0tz8czaa-cw18bq

Specific Values (4)

https://wolfram.com/xid/0tz8czaa-bmqd0y


https://wolfram.com/xid/0tz8czaa-e5asej

Evaluate for symbolic parameters:

https://wolfram.com/xid/0tz8czaa-nww7l


https://wolfram.com/xid/0tz8czaa-jvmxkv


https://wolfram.com/xid/0tz8czaa-ddt839

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

https://wolfram.com/xid/0tz8czaa-f2hrld


https://wolfram.com/xid/0tz8czaa-b1v4fb

Visualization (4)
Plot the UnitStep function:

https://wolfram.com/xid/0tz8czaa-ecj8m7

Visualize shifted UnitStep functions:

https://wolfram.com/xid/0tz8czaa-urg5gs

Visualize the composition of UnitStep with a periodic function:

https://wolfram.com/xid/0tz8czaa-lwo8te

Plot UnitStep in three dimensions:

https://wolfram.com/xid/0tz8czaa-i75zi3

Function Properties (10)
Function domain of UnitStep:

https://wolfram.com/xid/0tz8czaa-cl7ele

It is restricted to real inputs:

https://wolfram.com/xid/0tz8czaa-shglmn

Function range of UnitStep:

https://wolfram.com/xid/0tz8czaa-hkh7p

UnitStep has a jump discontinuity at the point :

https://wolfram.com/xid/0tz8czaa-5tj4t3

UnitStep is not an analytic function:

https://wolfram.com/xid/0tz8czaa-h5x4l2

It has both singularities and discontinuities:

https://wolfram.com/xid/0tz8czaa-mdtl3h


https://wolfram.com/xid/0tz8czaa-mn5jws

UnitStep is nondecreasing:

https://wolfram.com/xid/0tz8czaa-nlz7s

UnitStep is not injective:

https://wolfram.com/xid/0tz8czaa-poz8g


https://wolfram.com/xid/0tz8czaa-ctca0g

UnitStep is not surjective:

https://wolfram.com/xid/0tz8czaa-cxk3a6


https://wolfram.com/xid/0tz8czaa-frlnsr

UnitStep is non-negative:

https://wolfram.com/xid/0tz8czaa-84dui

UnitStep is neither convex nor concave:

https://wolfram.com/xid/0tz8czaa-8kku21

TraditionalForm formatting:

https://wolfram.com/xid/0tz8czaa-grt17d

Differentiation and Integration (6)
First derivative with respect to x:

https://wolfram.com/xid/0tz8czaa-krpoah

All higher-order derivatives the same:

https://wolfram.com/xid/0tz8czaa-xtz7ns

First derivative with respect to z:

https://wolfram.com/xid/0tz8czaa-bbvsao

Compute the indefinite integral using Integrate:

https://wolfram.com/xid/0tz8czaa-bponid

Verify the anti-derivative away from the singular point:

https://wolfram.com/xid/0tz8czaa-op9yly


https://wolfram.com/xid/0tz8czaa-bfdh5d

Integral over an infinite domain:

https://wolfram.com/xid/0tz8czaa-wyokxh

Integral Transforms (4)

https://wolfram.com/xid/0tz8czaa-cxb66w


https://wolfram.com/xid/0tz8czaa-kx3bqq


https://wolfram.com/xid/0tz8czaa-etyf41

Find the LaplaceTransform of UnitStep:

https://wolfram.com/xid/0tz8czaa-hnglci


https://wolfram.com/xid/0tz8czaa-bqb1e4

The convolution of UnitStep with itself:

https://wolfram.com/xid/0tz8czaa-bcycw


https://wolfram.com/xid/0tz8czaa-ic6l4j

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

https://wolfram.com/xid/0tz8czaa-bl9mbg

Compute a step response for a continuous-time system:

https://wolfram.com/xid/0tz8czaa-b6ajla


https://wolfram.com/xid/0tz8czaa-jywnb7


https://wolfram.com/xid/0tz8czaa-eokzo4


https://wolfram.com/xid/0tz8czaa-b0apsy

Compute a step response for a discrete-time system:

https://wolfram.com/xid/0tz8czaa-vcj3r


https://wolfram.com/xid/0tz8czaa-306u0


https://wolfram.com/xid/0tz8czaa-beulnm


https://wolfram.com/xid/0tz8czaa-midgnt

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

https://wolfram.com/xid/0tz8czaa-hfg2w

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

https://wolfram.com/xid/0tz8czaa-b3745d

Here is the resulting probability plotted:

https://wolfram.com/xid/0tz8czaa-m4brqx

Construct the Walsh function:

https://wolfram.com/xid/0tz8czaa-ba3nl3

https://wolfram.com/xid/0tz8czaa-ec02yw

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

https://wolfram.com/xid/0tz8czaa-d2ecit

https://wolfram.com/xid/0tz8czaa-czafsa

https://wolfram.com/xid/0tz8czaa-7c8gzi

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

https://wolfram.com/xid/0tz8czaa-bbc3ez

Properties & Relations (4)Properties of the function, and connections to other functions
The derivative of UnitStep is a piecewise function:

https://wolfram.com/xid/0tz8czaa-72k7v9

The derivative of HeavisideTheta is a distribution:

https://wolfram.com/xid/0tz8czaa-1gxedf

Expand into UnitStep of linear factors:

https://wolfram.com/xid/0tz8czaa-elqwm3


https://wolfram.com/xid/0tz8czaa-nm16rj

Convert into Piecewise:

https://wolfram.com/xid/0tz8czaa-xl91


https://wolfram.com/xid/0tz8czaa-memcqs

Integrate over finite and infinite domains:

https://wolfram.com/xid/0tz8czaa-cgtgxt


https://wolfram.com/xid/0tz8czaa-l87pyw

Possible Issues (3)Common pitfalls and unexpected behavior
Symbolic preprocessing of functions containing UnitStep can be time‐consuming:

https://wolfram.com/xid/0tz8czaa-e75zm1


https://wolfram.com/xid/0tz8czaa-iqa29

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

https://wolfram.com/xid/0tz8czaa-dw8hdm


https://wolfram.com/xid/0tz8czaa-yeaef

Differentiating Abs does not yield UnitStep:

https://wolfram.com/xid/0tz8czaa-foukrv

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

https://wolfram.com/xid/0tz8czaa-ucyzyp

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

https://wolfram.com/xid/0tz8czaa-x1v5fo

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
]}
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
]}