140:
485:; often it is also required that they are pairwise disjoint, i.e. form a partition of the domain. In order for the overall function to be called "piecewise", the subdomains are usually required to be intervals (some may be degenerated intervals, i.e. single points or unbounded intervals). For bounded intervals, the number of subdomains is required to be finite, for unbounded intervals it is often only required to be locally finite. For example, consider the piecewise definition of the
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333:{\displaystyle f(x)=\left\{{\begin{array}{lll}-3-x&{\text{if}}&x\leq -3\\x+3&{\text{if}}&-3\leq x\leq 0\\3-2x&{\text{if}}&0\leq x\leq 3\\0.5x-4.5&{\text{if}}&3\leq x\\\end{array}}\right.}
885:
In order to evaluate a piecewise-defined function at a given input value, the appropriate subdomain needs to be chosen in order to select the correct sub-function—and produce the correct output value.
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379:("subdomains") on which the function may be defined differently. Piecewise definition is actually a way of specifying the function, rather than a characteristic of the resulting function itself.
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Some sources only examine the function definition, while others acknowledge the property iff the function admits a partition into a piecewise definition that meets the conditions.
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For a piecewise-defined function to be differentiable on a given interval in its domain, the following conditions have to fulfilled in addition to those for continuity above:
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The pictured function, for example, is piecewise-continuous throughout its subdomains, but is not continuous on the entire domain, as it contains a jump discontinuity at
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912:, a function composed of polynomial sub-functions, possessing a high degree of smoothness at the places where the polynomial pieces connect
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1056:{\displaystyle f(x)={\begin{cases}\exp \left(-{\frac {1}{1-x^{2}}}\right),&x\in (-1,1)\\0,&{\text{otherwise}}\end{cases}}}
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at the points where two subintervals touch, the corresponding one-sided derivatives of the two neighboring subintervals coincide.
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have been used as a representation system to provide sparse approximations of this model class in 2D and 3D.
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In applied mathematical analysis, "piecewise-regular" functions have been found to be consistent with many
640:) is used, which negates the sign of the input value, making negative numbers positive. For all values of
1325:, where images are perceived at a first stage as consisting of smooth regions separated by edges (as in a
587:{\displaystyle |x|={\begin{cases}-x,&{\text{if }}x<0\\+x,&{\text{if }}x\geq 0.\end{cases}}}
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1292:. The filled circle indicates that the value of the right sub-function is used in this position.
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Kutyniok, Gitta; Lim, Wang-Q (2011). "Compactly supported shearlets are optimally sparse".
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A feasible weaker requirement is that all definitions agree on intersecting subdomains.
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function, smooth except for the existence of discontinuity curves. In particular,
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Piecewise defined functions are also commonly used for interpolation, such as in
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1507:"Differentiability of Piecewise Defined Functions – AP Central | College Board"
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there is no discontinuity at an endpoint of any subdomain within that interval.
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The following table documents the absolute value function at certain values of
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its sub-functions are continuous on the corresponding intervals (subdomains),
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on a given interval in its domain if the following conditions are met:
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Continuity and differentiability of piecewise-defined functions
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is used, which evaluates trivially to the input value itself.
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1306:
the one-sided derivatives exist at all intervals' endpoints,
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1549:"The set of continuous piecewise differentiable functions"
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its sub-functions are differentiable on the corresponding
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greater than or equal to zero, the second sub-function
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is rarely omitted at the start of the right column.
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Piecewise functions can be defined using the common
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906:, a function composed of power-law sub-functions
30:"Piecewise" redirects here. For other uses, see
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1529:
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481:The subdomains together must cover the whole
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1068:. These are infinitely differentiable, but
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27:Function defined by multiple sub-functions
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127:Learn how and when to remove this message
1547:Sofronidis, Nikolaos Efstathiou (2005).
1080:
617:less than zero, the first sub-function (
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138:
1533:A Course Of Mathematical Analysis Vol 1
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391:Graph of the absolute value function,
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65:adding citations to reliable sources
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1375:- a generalization of the concept
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1250:A piecewise-defined function is
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1610:Journal of Approximation Theory
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52:needs additional citations for
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471:{\displaystyle {\text{for}}}
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1583:"Introduction to shearlets"
1581:; Labate, Demetrio (2012).
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1239:{\displaystyle x_{0}=0.707}
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449:{\displaystyle {\text{if}}}
383:Notation and interpretation
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1511:apcentral.collegeboard.org
353:piecewise-defined function
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1632:10.1016/j.jat.2011.06.005
897:, a function composed of
895:Piecewise linear function
361:function defined by cases
145:piecewise linear function
1530:S. M. Nikolsky (1977).
1652:Functions and mappings
1553:Real Analysis Exchange
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1446:"Piecewise functions"
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1394:"Piecewise Functions"
1331:cartoon-like function
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1285:{\displaystyle x_{0}}
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61:improve this article
18:Piecewise continuous
1472:Weisstein, Eric W.
1420:Weisstein, Eric W.
432:functional notation
1398:www.mathsisfun.com
1373:Piecewise property
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32:Piecewise property
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899:line segments
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72:Find sources:
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50:This article
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1559:(1): 13–22.
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1514:. Retrieved
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1481:. Retrieved
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1453:. Retrieved
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1429:. Retrieved
1425:
1401:. Retrieved
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1317:Applications
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143:Plot of the
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114:
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59:Please help
54:verification
51:
1070:analyticity
373:partitioned
345:mathematics
1594:Birkhäuser
1516:2024-08-26
1483:2024-07-20
1455:2020-09-29
1431:2020-08-24
1403:2020-08-24
1380:References
1303:intervals,
1252:continuous
489:function:
117:March 2017
87:newspapers
1623:1002.2661
1598:Here: p.8
1590:Shearlets
1565:0147-1937
1366:Piecewise
1339:shearlets
1191:≤
1163:−
1154:−
1043:otherwise
1015:−
1009:∈
980:−
968:−
960:
778:−
746:−
625:−
572:≥
524:−
377:intervals
318:≤
300:−
284:≤
278:≤
257:−
244:≤
238:≤
232:−
204:−
201:≤
183:−
177:−
1646:Category
1352:See also
916:B-spline
889:Examples
565:if
536:if
365:function
1596:: 1–38.
1327:cartoon
363:) is a
359:, or a
101:scholar
1563:
910:Spline
483:domain
369:domain
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