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In mathematics, a piecewise linear function

where V is a vector space and is a subset of a vector space, is any function with the property that 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 . Then f is piecewise linear if and only if 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 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.

1 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 spaceEuclidean space is the usual n dimensional mathematical space, a generalization of the 2- and 3-dimensional spaces studied by Euclid. Formally, for any non-negative integer n n dimensional Euclidean space is the set R n (where R is the set of real numbers.

2 See also

PiecewiseIn mathematics, a function f ''x of a real number variable x is defined piecewise if f ''x is given by different expressions on various intervals. An example is given by the Heaviside step function. According to the standard definitions, this is a single

Real analysisReal analysis is that branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. It can be seen as a rigorous version of calculus and studies concepts such as sequences and their limits, continuity, differentiation Geometric topology

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Topics: Piecewise Linear Function Functions Space Convex Geometric...

PiecewiseIn mathematics, a function f ''x of a real number variable x is defined piecewise if f ''x is given by different expressions on various intervals. An example is given by the Heaviside step function. According to the standard definitions, this is a single

Piecewise linearIn mathematics, a piecewise linear function :, where V is a vector space and is a subset of a vector space, is any function with the property that can be decomposed into finitely many convex polytopes, such that f is equal to a linear function on each of

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