UnitTriangle: Unit triangle function

UnitTriangle: Unit triangle function—Wolfram Documentation

Differentiation and Integration

UnitTriangle[x]

represents the unit triangle function on the interval .

UnitTriangle[x₁,x₂,…]

represents the multidimensional unit triangle function on the interval .

Details

UnitTriangle[x] is equivalent to Piecewise[{{x+1,-1≤x<0},{1-x,0≤x≤1}}].
UnitTriangle can be used in integrals and integral transforms.
UnitTriangle has attribute Orderless.
UnitTriangle automatically threads over lists. »

Examples

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Basic Examples (4)

Evaluate numerically:

Plot in one dimension:

Plot in two dimensions:

UnitTriangle is a piecewise function:

Scope (36)

Numerical Evaluation (7)

Evaluate numerically:

Evaluate to high precision:

For inputs between -1 and 1, the precision of the output tracks the precision of the input:

For inputs outside that range, the result is exact:

Evaluate efficiently at high precision:

UnitTriangle threads over lists:

Compute the elementwise values of an array using automatic threading:

Or compute the matrix UnitTriangle function using MatrixFunction:

Compute average-case statistical intervals using Around:

Specific Values (4)

Values of UnitTriangle at fixed points:

Value at zero:

Evaluate symbolically:

Find a value of x for which UnitTriangle[x]=0.4:

Visualization (4)

Function Properties (11)

Differentiation and Integration (6)

First derivative with respect to x:

Higher-order derivatives with respect to x:

First derivative with respect to z:

Series expansion at the origin:

Compute the indefinite integral using Integrate:

Verify the anti-derivative away from the singular points:

Definite integral:

Integral Transforms (4)

Applications (4)

Integrate a piecewise function involving UnitTriangle symbolically and numerically:

Solve a differential equation involving UnitBox and UnitTriangle:

Visualize discontinuities in the wavelet domain:

Detail coefficients in the region of discontinuities have larger values:

Generate data from some distribution:

Apply mean shift until all data points have converged:

Gather the result into clusters:

Visualize the clustering:

Properties & Relations (4)

Wolfram Research (2008), UnitTriangle, Wolfram Language function, https://reference.wolfram.com/language/ref/UnitTriangle.html.

Text

Wolfram Research (2008), UnitTriangle, Wolfram Language function, https://reference.wolfram.com/language/ref/UnitTriangle.html.

CMS

Wolfram Language. 2008. "UnitTriangle." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/UnitTriangle.html.

APA

Wolfram Language. (2008). UnitTriangle. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/UnitTriangle.html

BibTeX

@misc{reference.wolfram_2025_unittriangle, author="Wolfram Research", title="{UnitTriangle}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/UnitTriangle.html}", note=[Accessed: 22-February-2026]}

BibLaTeX

@online{reference.wolfram_2025_unittriangle, organization={Wolfram Research}, title={UnitTriangle}, year={2008}, url={https://reference.wolfram.com/language/ref/UnitTriangle.html}, note=[Accessed: 22-February-2026]}