UnitTriangle

The unit triangle function: 1 - |x| for -1 <= x <= 1, 0 otherwise.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageUnitTriangle✓↗

UnitTriangle(x)max(1 - |x|, 0).

Typed signatures

Type	Package	Applies
(real) -> number	analytic	always

Domain: Elementary functions

Details

• Continuous everywhere, including the endpoints -- UnitTriangle(±1) = 0 falls out of the formula itself, no special case needed.
• The same function as HeavisideLambda; Wolfram keeps both names for the two contexts (elementary vs. the Heaviside/Dirac generalized-function family) it shows up in.
• N(x, d) past a double's digits keeps the bignum operand and returns $1 - |x|$ in its digits.

Examples

1 more kept as data, checked against the oracles

Basic #

The peak

live ↗

In$\mathrm{UnitTriangle}(0)$Out$1$

Zero at the endpoint

live ↗

In$\mathrm{UnitTriangle}(1)$Out$0$

live ↗

In$\mathrm{UnitTriangle}(-1)$Out$0$

UnitTriangle as a two-branch Piecewise

live ↗

In$\mathrm{PiecewiseExpand}(\mathrm{UnitTriangle}(x))$Out$\mathrm{Piecewise}(\bigl\lbrack\bigl\lbrack1-x, 0≤x≤1\bigr\rbrack, \bigl\lbrack x+1, -1≤x∧x\lt0\bigr\rbrack\bigr\rbrack, 0)$

Scope #

Zero past the endpoints

live ↗

In$\mathrm{UnitTriangle}(2)$Out$0$

Exact rational input, exact rational output

live ↗

In$\mathrm{UnitTriangle}(\frac{1}{3})$Out$\frac{2}{3}$

$1-|x|=\frac34$ at $x=\frac14$

live ↗

In$\mathrm{UnitTriangle}(\frac{1}{4})$Out$\frac{3}{4}$

The triangle is cut to what $0 < x < 2$ leaves open

live ↗

In$\mathrm{PiecewiseExpand}(\mathrm{UnitTriangle}(x), 0\lt x\lt2)$Out$\mathrm{Piecewise}(\bigl\lbrack\bigl\lbrack1-x, x≤1\bigr\rbrack\bigr\rbrack, 0)$

live ↗

In$N(\mathrm{UnitTriangle}(-\frac{1}{7}), 50)$Out$0.\overline{857142}$

live ↗

In$\operatorname{Normal}(\operatorname{Series}(\mathrm{UnitTriangle}(x), x, 0, 3))$Out$\frac{1}{6}(\mathrm{UnitTriangle}^{\prime\prime\prime}(0)x^3)+\frac{1}{2}(\mathrm{UnitTriangle}^{\prime\prime}(0)x^2)+x\mathrm{UnitTriangle}^{\prime}(0)+1$

Properties #

The same function as HeavisideLambda

live ↗

In$\mathrm{UnitTriangle}(\frac{1}{4})=\mathrm{HeavisideLambda}(\frac{1}{4})$Out$\mathrm{True}$

About

Examples	12
Checked against	Wolfram Language 11 agree1 unevaluated
Crosswalk	Wolfram Language