Editing Piecewise linear function (section)

==Generalizations==
[[Image:Piecewise linear function2D.svg|right|thumbnail|A piecewise linear function of two arguments (top) and the convex polytopes on which it is linear (bottom)]]

The notion of a piecewise linear function makes sense in several different contexts. Piecewise linear functions may be defined on [[dimension|''n''-dimensional]] [[Euclidean space]], or more generally any [[vector space]] or [[affine space]], as well as on [[piecewise linear manifold]]s and [[simplicial complex]]es (see [[simplicial map]]). In each case, the function may be [[real number|real]]-valued, or it may take values from a vector space, an affine space, a piecewise linear manifold, or a simplicial complex. (In these contexts, the term “linear” does not refer solely to [[linear map|linear transformations]], but to more general [[affine transformation|affine linear]] functions.)

In dimensions higher than one, it is common to require the domain of each piece to be a [[polygon]] or [[polytope]]. This guarantees that the graph of the function will be composed of polygonal or polytopal pieces.

[[Spline (mathematics)|Splines]] generalize piecewise linear functions to higher-order polynomials, which are in turn contained in the category of piecewise-differentiable functions, [[PDIFF]].