Piecewise linear function

In mathematics, a piecewise linear function

:$f:\; Omega\; o\; V$,

where "V" is a vector space and $Omega$ is asubset 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 linearif 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" + "bI".}

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.

ee also

*Piecewise
*Piecewise linear manifold