Piecewise linear function

A function (blue) and a piecewise linear approximation to it (red).

A piecewise linear function in two dimensions (top) and the convex polytopes on which it is linear (bottom).

In mathematics, a piecewise linear function

$f:\Omega \to V$,

where V is a vector space and $\Omega$ is a subset of a vector space, is any function with the property that $\Omega$ can be decomposed into finitely many convex polytopes, such that f is equal to a linear function on each of these polytopes. (Here, the term linear function is not restricted to linear transformations, but is used in the more general sense of affine transformation.)

A special case is when f is a real-valued function on an interval $[x_{1},x_{2}]$. Then f is piecewise linear if and only if $[x_{1},x_{2}]$ can be partitioned into finitely many sub-intervals, such that on each such sub-interval I, f is equal to a linear function

f(x) = a_Ix + b_I.

The absolute value function $f(x)=|x|$ is a good example of a piecewise linear function. Other examples include the square wave, the sawtooth function, and the floor function.

Important sub-classes of piecewise linear functions include the continuous piecewise linear functions and the convex piecewise linear functions. Splines generalize piecewise linear functions to higher-order polynomials.

See also

• Piecewise
• Piecewise linear manifold