In mathematics, a piecewise linear function is a piecewisedefined function whose pieces are linear.
The function defined by:
is piecewise linear with four pieces. (The graph of this function is shown to the right.) Since the graph of a linear function is a line, the graph of a piecewise linear function consists of line segments and rays. If the function is continuous, the graph will be a polygonal curve.
Other examples of piecewise linear functions include the absolute value function, the square wave, the sawtooth function, and the floor function.
The notion of a piece wise linear function makes sense in several different contexts. Piecewise linear functions may be defined on ndimensional Euclidean space, or more generally any vector space or affine space, as well as on piecewise linear manifolds, simplicial complexes, and so forth. In each case, the function may be realvalued, or it may take values from a vector space, an affine space, a piecewiselinear manifold, or a simplicial complex. (In these contexts, the term “linear” does not refer solely to linear transformations, but to more general 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.
Important subclasses of piecewise linear functions include the continuous piecewise linear functions and the convex piecewise linear functions. Splines generalize piecewise linear functions to higherorder polynomials, and more one can speak of piecewisedifferentiable functions, as in PDIFF.
