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A piecewise linear function is a function defined on a (possibly unbounded) interval of real numbers, such that there is a collection of intervals on each of which the function is an affine function. (Thus "piecewise linear" is actually defined to mean "piecewise affine ".)
Terms like piecewise linear, piecewise smooth, piecewise continuous, and others are very common. The meaning of a function being piecewise P {\displaystyle P} , for a property P {\displaystyle P} is roughly that the domain of the function can be partitioned into pieces on which the property P {\displaystyle P} holds, but is used slightly ...
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
These may be defined as indeed higher-dimensional piecewise linear functions (see second figure below). Example of bilinear interpolation on the unit square with the z values 0, 1, 1, and 0.5 as indicated. Interpolated values in between are represented by colour. A piecewise linear function in two dimensions (top) and the convex polytopes on ...
In mathematics, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted ⌊x⌋ or floor(x). Similarly, the ceiling function maps x to the least integer greater than or equal to x , denoted ⌈ x ⌉ or ceil( x ) .
Piecewise linear curve, a connected sequence of line segments; Piecewise linear function, a function whose domain can be decomposed into pieces on which the function is linear; Piecewise linear manifold, a topological space formed by gluing together flat spaces; Piecewise linear homeomorphism, a topological equivalence between two piecewise ...
When =, in particular, () approaches the linear piecewise polynomials, i.e. connecting the interpolation points with straight lines. The role played by p {\displaystyle p} in the process of minimizing V p {\displaystyle V_{p}} is to control the importance of the size of the fluctuations away from the mean value.
Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function. Kronecker delta function: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise.