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PIECEWISE LINEAR
In mathematics, a piecewise linear function
 ,
where V is a vector space and Ω is a subset of a vector space, is any function with the property that Ω can be decomposed into finitely many convex polytopes, such that f is equal to a linear function on each of these polytopes.
A special case is when f is a real-valued function on an interval [x1,x2]. Then f is piecewise linear if and only if [x1,x2] 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) = aIx + 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.
PL manifolds
The idea of a piecewise linear (PL) structure on a topological manifold M is used in geometric topology. Smooth manifolds have PL structures, but not conversely, in general. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear functions. A more slick definition is to use a sheaf, locally isomorphic to the sheaf of piecewise linear functions on Euclidean space.
See also
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