Piecewise
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![](http://upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Piecewise_linear_function_gnuplot.svg/280px-Piecewise_linear_function_gnuplot.svg.png)
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function whose domain is partitioned into several intervals ("subdomains") on which the function may be defined differently.[1][2][3] Piecewise definition is actually a way of specifying the function, rather than a characteristic of the resulting function itself.
A function property holds piecewise for a function, if the function can be piecewise defined in a way that the property holds for every subdomain. Examples of functions with such piecewise properties are
Notation and interpretation
![](http://upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Absolute_value.svg/280px-Absolute_value.svg.png)
Piecewise functions can be defined using the common
For all values of less than zero, the first sub-function () is used, which negates the sign of the input value, making negative numbers positive. For all values of greater than or equal to zero, the second sub-function () is used, which evaluates trivially to the input value itself.
The following table documents the absolute value function at certain values of :
x | f(x) | Sub-function used |
---|---|---|
−3 | 3 | |
−0.1 | 0.1 | |
0 | 0 | |
1/2 | 1/2 | |
5 | 5 |
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.
Examples
- Piecewise linear function, a function composed of line segments
- Step function, a function composed of constant sub-functions
- Absolute value[2]
- Triangular function
- Broken power law, a function composed of power-law sub-functions
- Spline, a function composed of polynomial sub-functions, possessing a high degree of smoothness at the places where the polynomial pieces connect
- PDIFF
and some other common Bump functions. These are infinitely differentiable, but analyticity holds only piecewise.
Continuity and differentiability of piecewise-defined functions
![](http://upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Upper_semi.svg/280px-Upper_semi.svg.png)
A piecewise-defined function is continuous on a given interval in its domain if the following conditions are met:
- its sub-functions are continuous on the corresponding intervals (subdomains),
- there is no discontinuity at an endpoint of any subdomain within that interval.
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 . The filled circle indicates that the value of the right sub-function is used in this position.
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:
- its sub-functions are differentiable on the corresponding open intervals,
- the one-sided derivatives exist at all intervals' endpoints,
- at the points where two subintervals touch, the corresponding one-sided derivatives of the two neighboring subintervals coincide.
Applications
In applied mathematical analysis, "piecewise-regular" functions have been found to be consistent with many models of the human visual system, where images are perceived at a first stage as consisting of smooth regions separated by edges.[5] In particular, shearlets have been used as a representation system to provide sparse approximations of this model class in 2D and 3D.
Piecewise defined functions are also commonly used for interpolation, such as in nearest-neighbor interpolation.
See also
![](http://upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png)
References
- ^ "Piecewise Functions". www.mathsisfun.com. Retrieved 2020-08-24.
- ^ a b c d Weisstein, Eric W. "Piecewise Function". mathworld.wolfram.com. Retrieved 2020-08-24.
- ^ "Piecewise functions". brilliant.org. Retrieved 2020-09-29.
- ^ A feasible weaker requirement is that all definitions agree on intersecting subdomains.
- ^ Kutyniok, Gitta; Labate, Demetrio (2012). "Introduction to shearlets" (PDF). Shearlets. Birkhäuser: 1–38. Here: p.8