Piecewise

Piecewise

In mathematics, a piecewise-defined function (also called a piecewise function) is a function which is defined by multiple subfunctions, each subfunction applying to a certain interval of the main function's domain (a subdomain). Piecewise is actually a way of expressing the function, rather than a characteristic of the function itself, but with additional qualification, it can describe the nature of the function. For example, a piecewise polynomial function: a function that is a polynomial on each of its subdomains, but possibly a different one on each.

The word piecewise is also used to describe any property of a piecewise-defined function that holds for each piece but may not hold for the whole domain of the function. A function is piecewise differentiable or piecewise continuously differentiable if each piece is differentiable throughout its subdomain, even though the whole function may not be differentiable at the points between the pieces. In convex analysis, the notion of a derivative may be replaced by that of the subderivative for piecewise functions. Although the "pieces" in a piecewise definition need not be intervals, a function isn't called "piecewise linear" or "piecewise continuous" or "piecewise differentiable" unless the pieces are intervals.

Read more about Piecewise:  Notation and Interpretation, Continuity

Other articles related to "piecewise":

Stallings–Zeeman Theorem - Statement of The Theorem
... the homotopy type of the m-dimensional sphere Sm and that M is locally piecewise linearly homeomorphic to m-dimensional Euclidean space Rm ... Then M is homeomorphic to Sm under a map that is piecewise linear except possibly at a single point x ... That is, M {x} is piecewise linearly homeomorphic to Rm ...
Triangulation (topology) - Piecewise Linear Structures
... there is a slightly stronger notion of triangulation a piecewise-linear triangulation (sometimes just called a triangulation) is a triangulation with the extra property that the ... For instance, in a two-dimensional piecewise-linear manifold formed by a set of vertices, edges, and triangles, the link of a vertex s consists of the cycle of vertices and edges ... to the n-sphere) with a triangulation that is not piecewise-linear it has a simplex whose link is the Poincaré homology sphere, a three-dimensional manifold ...
Piecewise - Continuity
... A piecewise function is continuous on a given interval if the following conditions are met it is defined throughout that interval its constituent ... The pictured function, for example, is piecewise continuous throughout its subdomains, but is not continuous on the entire domain ...
PDIFF
... In geometric topology, PDIFF, for piecewise differentiable, is the category of piecewise-smooth manifolds and piecewise-smooth maps between them ... between them – and PL – the category of piecewise linear manifolds and piecewise linear maps between them – and the reason it is defined is to allow one to ... Further, piecewise functions such as splines and polygonal chains are common in mathematics, and PDIFF provides a category for discussing them ...
Piecewise Syndetic Set - Properties
... A set is piecewise syndetic if and only if it is the intersection of a syndetic set and a thick set ... If S is piecewise syndetic then S contains arbitrarily long arithmetic progressions ... A set S is piecewise syndetic if and only if there exists some ultrafilter U which contains S and U is in the smallest two-sided ideal of, the Stone–Čech compactification of the natural numbers ...