Editing Piecewise function (section)

{{Short description|Function defined by multiple sub-functions}}
{{Redirect|Piecewise||Piecewise property (disambiguation){{!}}Piecewise property}}
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[[File:Piecewise linear function gnuplot.svg|thumb|280px|Plot of the [[piecewise linear function]] <math>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.</math>]]

In [[mathematics]], a '''piecewise function''' (also called a '''piecewise-defined function''', a '''hybrid function''', or a '''function defined by cases''') is a [[function (mathematics)|function]] whose [[Domain of a function|domain]] is [[partition of a set|partitioned]] into several [[interval (mathematics)|intervals]] ("subdomains") on which the function may be defined differently.<ref>{{Cite web|title=Piecewise Functions|url=https://www.mathsisfun.com/sets/functions-piecewise.html|access-date=2020-08-24|website=www.mathsisfun.com}}</ref><ref name=mathworld>{{mathworld|title=Piecewise Function|id=PiecewiseFunction}}</ref><ref>{{Cite web|title=Piecewise functions|url=https://brilliant.org/wiki/piecewise-functions/|access-date=2020-09-29|website=brilliant.org}}</ref> Piecewise definition is actually a way of specifying the function, rather than a characteristic of the resulting function itself, as every function whose domain contains at least two points can be rewritten as a piecewise function. The first three paragraphs of this article only deal with this first meaning of "piecewise".

Terms like '''piecewise linear''', '''piecewise smooth''', '''piecewise continuous''', and others are also very common. The meaning of a function being piecewise <math>P</math>, for a property <math>P</math> is roughly that the domain of the function can be partitioned into pieces on which the property <math>P</math> holds, but is used slightly differently by different authors.<ref>{{Cite book |last=S. M. Nikolsky |url=https://archive.org/details/nikolsky-a-course-of-mathematical-analysis-vol-1-mir |title=A Course Of Mathematical Analysis Vol 1 |year=1977 |pages=178}}</ref><ref>{{Cite journal |last=Sofronidis |first=Nikolaos Efstathiou |date=2005 |title=The set of continuous piecewise differentiable functions. |url=https://projecteuclid.org/journals/real-analysis-exchange/volume-31/issue-1/The-set-of-continuous-piecewise-differentiable-functions/rae/1149516810.full |journal=Real Analysis Exchange |volume=31 |issue=1 |pages=13–22 |doi=10.14321/realanalexch.31.1.0013 |issn=0147-1937}}</ref> Unlike the first meaning, this is a property of the function itself and not only a way to specify it. Sometimes the term is used in a more global sense involving triangulations; see [[Piecewise linear manifold]].