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In mathematics, a piecewise function (also called a piecewise-defined function, a hybrid function, or a function defined 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 ...
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: Piecewise constant function, also known as a step function; Piecewise linear function; Piecewise continuous function
Since the graph of an affine(*) function is a line, the graph of a piecewise linear function consists of line segments and rays. The x values (in the above example −3, 0, and 3) where the slope changes are typically called breakpoints, changepoints, threshold values or knots.
Plot of Weierstrass function over the interval [−2, 2]. Like some other fractals , the function exhibits self-similarity : every zoom (red circle) is similar to the global plot. In mathematics , the Weierstrass function , named after its discoverer, Karl Weierstrass , is an example of a real-valued function that is continuous everywhere but ...
Single knots at 1/3 and 2/3 establish a spline of three cubic polynomials meeting with C 2 parametric continuity. Triple knots at both ends of the interval ensure that the curve interpolates the end points. In mathematics, a spline is a function defined piecewise by polynomials.
For an infinite set of functions, the same notions may be defined using the infimum in place of the minimum, and the supremum in place of the maximum. [ 1 ] For continuous functions from a given class, the lower or upper envelope is a piecewise function whose pieces are from the same class.
A valuation V is called piecewise-constant, if the corresponding value-density function v is a piecewise-constant function. In other words: there is a partition of the resource C into finitely many regions, C 1 ,..., C k , such that for each j in 1,..., k , the function v inside C j equals some constant U j .
The integral of a positive real function f between boundaries a and b can be interpreted as the area under the graph of f, between a and b.This notion of area fits some functions, mainly piecewise continuous functions, including elementary functions, for example polynomials.