Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate efficiently at high precision:

TriangleWave threads over lists in the last argument:

Compute the elementwise values of an array using automatic threading:

Or compute the matrix TriangleWave function using MatrixFunction:

Value at zero:

Values of TriangleWave at fixed points:

Evaluate symbolically:

Find a value of for which the TriangleWave[x]=0.5:

Plot the TriangleWave function:

Visualize scaled TriangleWave functions:

Visualize TriangleWave functions with different maximum and minimum values:

Plot TriangleWave in three dimensions:

Function domain of TriangleWave:

It is restricted to real inputs:

Function range of TriangleWave[x]:

TriangleWave is periodic with period 1:

TriangleWave is an odd function:

The area under one period is zero:

TriangleWave is not an analytic function because it is singular at the half-integers:

However, it is continuous:

TriangleWave[x] is neither nondecreasing nor nonincreasing:

TriangleWave is not injective:

TriangleWave[x] is not surjective:

TriangleWave[x] is neither non-negative nor non-positive:

TriangleWave is neither convex nor concave:

First derivative with respect to :

Derivative of the two-argument form with respect to :

The second (and higher) derivatives are zero except at points where the derivative does not exist:

If a==b, TriangleWave[{a,b},x] is constant and its derivatives are zero everywhere:

Integrals over finite domains:

FourierSeries:

Since TriangleWave is odd, FourierTrigSeries gives a simpler result:

The two results are equivalent:

FourierCosSeries of a scaled TriangleWave:

Maclaurin series:

Series expansion at a singular point:

Taylor expansion at a generic point: