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Function defined by multiple sub-functions
mathematics, a piecewise function (also called a piecewise-defined function, a hybrid function, or a function defined by cases) is a function whose domain
Piecewise_function
Type of mathematical function
piecewise linear or segmented function is a real-valued function of a real variable, whose graph is composed of straight-line segments. A piecewise linear
Piecewise_linear_function
Linear combination of indicator functions of real intervals
intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces. A function f : R → R {\displaystyle f\colon
Step_function
Piecewise function that clamps its input to be non-negative
the ramp function. The ramp function (R(x) : R → R0+) may be defined analytically in several ways. Possible definitions are: A piecewise function: R ( x
Ramp_function
Type of mathematical function
elementary, such as piecewise-defined functions. More generally, in some modern treatments, elementary functions comprise the set of functions previously enumerated
Elementary_function
Indicator function of positive numbers
and represented the function as 1. Taking the convention that H(0) = 1, the Heaviside function may be defined as: A piecewise function: H ( x ) := { 1 ,
Heaviside_step_function
Function returning minus 1, zero or plus 1
{\displaystyle \operatorname {sgn}(x)} . The signum function of a real number x {\displaystyle x} is a piecewise function which is defined as follows: sgn x :=
Sign_function
Mathematical function defined piecewise by polynomials
In mathematics, a spline is a function defined piecewise by polynomials. In interpolating problems, spline interpolation is often preferred to polynomial
Spline_(mathematics)
Set of all things that may be the input of a mathematical function
∈ R : x ≠ 0 } {\displaystyle \{x\in \mathbb {R} :x\neq 0\}} . The piecewise function f {\displaystyle f} defined by f ( x ) = { 1 / x x ≠ 0 0 x = 0 , {\displaystyle
Domain_of_a_function
Function used in signal processing
astronomy, inspired by the Planck distribution. It is defined as a piecewise function: w [ 0 ] = 0 , w [ n ] = ( 1 + exp ( ε N n − ε N ε N − n ) ) − 1
Window_function
Tent function, often used in signal processing
sometimes called the Bartlett window. The most common definition is as a piecewise function: tri ( x ) = Λ ( x ) = def max ( 1 − | x | , 0 ) = { 1 − | x
Triangular_function
Function whose graph is 0, then 1, then 0 again, in an almost-everywhere continuous way
convolution of the dis-continuous rectangular function results in the triangular function, a piecewise defined spline that is continuous, but not continuously
Rectangular_function
Topics referred to by the same term
mathematics, piecewise properties may be exemplified in: A function property holds piecewise for a function, if the function can be piecewise-defined in
Piecewise_property
Function that is continuous everywhere but differentiable nowhere
the cosine function can be replaced in the infinite series by a piecewise linear "zigzag" function. G. H. Hardy showed that the function of the above
Weierstrass function (nowhere-differentiable function)
Weierstrass_function_(nowhere-differentiable_function)
Browser-based graphing calculator
graph restriction, simultaneous graphing, piecewise function graphing, recursive function graphing, polar function graphing, two types of graphing grids –
Desmos
Generalized function whose value is zero everywhere except at zero
numerical analysis, a piecewise linear approximation to the identity is desirable. This can be obtained by taking η1 to be a hat function. With this choice
Dirac_delta_function
function (e.g. arcsine is the inverse of sine). Implicit function: defined implicitly by a relation between the argument(s) and the value. Piecewise function:
List_of_types_of_functions
Linear map or polynomial function of degree one
Geometrically, the graph of the function must pass through the origin. Homogeneous function Nonlinear system Piecewise linear function Linear approximation Linear
Linear_function
to 1, but using simpler functions, especially piecewise linear functions or piecewise constant functions. These are preferred where speed of computation
Hard_sigmoid
Polynomial function of degree 3
is a piecewise cubic function. If the value of a function is known at several points, cubic interpolation consists in approximating the function by a
Cubic_function
Multivalued function in mathematics
W function by piecewise minimax rational function approximation with variable transformation. doi:10.13140/RG.2.2.30264.37128. "Lambert W Functions -
Lambert_W_function
Method of curve fitting
higher-dimensional piecewise linear functions (see second figure below). Many libraries and shading languages have a "lerp" helper-function (in GLSL known
Linear_interpolation
Generalization of the Riemann integral
C=\{c_{i}:i\in \mathbb {N} \}} can be calculated. Consider for example the piecewise function f ( t ) = { 0 , if t ∈ [ 0 , 1 ] and rational, 1 , if t ∈ [ 0
Henstock–Kurzweil_integral
Punctuation mark (;)
Archived from the original on 28 August 2019. Weisstein, Eric W. "Piecewise Function". Wolfram MathWorld. Retrieved 2024-07-20. Steyer, Carly (7 July 2016)
Semicolon
Numerical method for solving physical or engineering problems
chooses basis functions. We used piecewise linear basis functions in our discussion, but it is common to use piecewise polynomial basis functions. Separate
Finite_element_method
Topics referred to by the same term
Piecewise linear may refer to: Piecewise linear curve, a connected sequence of line segments Piecewise linear function, a function whose domain can be
Piecewise_linear
Topological manifold with a piecewise linear structure on it
an atlas, such that one can pass from chart to chart in it by piecewise linear functions. This is slightly stronger than the topological notion of a triangulation
Piecewise_linear_manifold
Degree of differentiability of a function or map
theory) Smoothing – Fitting an approximating function to data Spline – Mathematical function defined piecewise by polynomials Sobolev mapping Warner, Frank
Smoothness
Mapping arbitrary data to fixed-size values
A hash function is any function that can be used to map data of arbitrary size to fixed-size values, though there are some hash functions that support
Hash_function
Loss function used in robust regression
The Huber loss function describes the penalty incurred by an estimation procedure f. Huber (1964) defines the loss function piecewise by L δ ( a ) = {
Huber_loss
Type of function in mathematics
nonempty open subset of C {\displaystyle \mathbb {C} } . Piecewise defined functions (functions given by different formulae in different regions) are typically
Analytic_function
Piece-wise division of objects
A piecewise-constant valuation is a kind of a function that represents the utility of an agent over a continuous resource, such as land. It occurs when
Piecewise-constant_valuation
Standard for HDTV image encoding and signal characteristics
analog-to-digital converters. Thus the standard opted for a piecewise function that combines a simple linear function V=aL for low light levels and a shifted power
Rec._709
Functional relationship between two quantities
logarithmic) The power law of forgetting A broken power law is a piecewise function, consisting of two or more power laws, combined with a threshold.
Power_law
Two continuous functions can be glued together to create another continuous function
continuous functions can be "glued together" to create another continuous function. The lemma is implicit in the use of piecewise functions. For example
Pasting_lemma
Traditional Chinese game of four players
modifiers apply (see above), with the point translation function being a piecewise function: a constant amount is given for scoreless hands, and the
Scoring_in_Mahjong
Algorithm used for pathfinding and graph traversal
(g(n)+h(n))2+4ϵ(ϵ−1)h(n)2 ]. Piecewise Upward/Downward Curve (pwXU/pwXD). Similar to XUP/XDP but with piecewise functions instead of parabola. Solution
A*_search_algorithm
functions or groups of functions are important enough to deserve their own names. This is a listing of articles which explain some of these functions
List of mathematical functions
List_of_mathematical_functions
Mathematical analysis of discontinuous points
valued function f of a real variable x, defined in a neighborhood of the point x0 at which f is discontinuous. Consider the piecewise function
Classification of discontinuities
Classification_of_discontinuities
continuous functions from a given class, the lower or upper envelope is a piecewise function whose pieces are from the same class. For functions of a single
Lower_envelope
Spline function
a fundamental building block for all spline functions of that degree. A B-spline is defined as a piecewise polynomial of order n {\displaystyle n} , meaning
B-spline
Distance from a point to the boundary of a set
Ω is a subset of the Euclidean space Rn with piecewise smooth boundary, then the signed distance function is differentiable almost everywhere, and its
Signed_distance_function
Integral transform and linear operator
be mitigated by use of a window function to taper h ~ [ n ] {\displaystyle {\tilde {h}}[n]} to zero. Method 2: Piecewise convolution. It is well known that
Hilbert_transform
Statistical method of dividing data into equal-sized intervals for analysis
x⌊h⌋, and x⌈h⌉. (For notation, see floor and ceiling functions). The first three are piecewise constant, changing abruptly at each data point, while
Quantile
Method for estimating new data within known data points
fit smoothly together. The resulting function is called a spline. For instance, the natural cubic spline is piecewise cubic and twice continuously differentiable
Interpolation
Unit of measurement
use of Bang measure could apply when re-engineering (either complete or piecewise) is being considered, as discussed in Maintenance of Operational Systems—An
Function_point
Process that changes pixel intensity
and visually appealing image. It can be described as the following piecewise function: I N = { s 1 r 1 I if I < r 1 s 2 − s 1 r 1 − r 2 ( I − r 1 ) if
Normalization (image processing)
Normalization_(image_processing)
Method for load calculation in construction
a piecewise manner, since in practice a load isn't typically a continuous function. Point loads can be modeled with help of the Dirac delta function. For
Euler–Bernoulli_beam_theory
Mathematical function with no sudden changes
discontinuities Coarse function Continuous function (set theory) Continuous stochastic process Normal function Open and closed maps Piecewise Squeeze mapping
Continuous_function
Strong form of uniform continuity
a metric on M or N. This structure is intermediate between that of a piecewise-linear manifold and a topological manifold: a PL structure gives rise
Lipschitz_continuity
Audio process
arctan, polynomial functions, or piecewise functions (such as the hard clipping function) are commonly used as waveshaping transfer functions. It is also possible
Waveshaper
Nearest integers from a number
\end{aligned}}} None of the functions discussed in this article are continuous, but all are piecewise linear: the functions ⌊ x ⌋ {\displaystyle \lfloor
Floor_and_ceiling_functions
Type of activation function
01–0.3. f(x)={xx>0,αxx≤0,f′(x)={1x>0,αx≤0. The same function can also be expressed without the piecewise notation as: f(x)=1+α2x+1−α2|x| Parametric ReLU (PReLU
Rectified_linear_unit
Phenomenon in nuclear engineering
for the neutron flux using a piecewise function solution to a differential equation involving a Heaviside step function in energy intervals of width (
Placzek_transient
Model of signal reconstruction in digital-to-analog (DAC) converters
instead output a sequence of rectangular pulses, xZOH(t) (a piecewise constant function), means that there is an inherent effect of the ZOH on the effective
Zero-order_hold
Set of functions between two fixed sets
functions B M O ( Ω ) {\displaystyle BMO(\Omega )} , space of bounded mean oscillation. Also called John–Nirenberg space linear functions piecewise linear
Function_space
Failure of convergence in interpolation
the behaviour of the Runge function. The method has demonstrated that it has a better interpolation performance than Piecewise polynomials (splines) to
Runge's_phenomenon
Oscillatory error in Fourier series
piecewise continuously differentiable periodic function around a jump discontinuity. The N {\textstyle N} th partial Fourier series of the function (formed
Gibbs_phenomenon
Relation among continuous functions
Mathematical term Normal function – Function of ordinals in mathematics Piecewise – Function defined by multiple sub-functionsPages displaying short descriptions
Equicontinuity
coordinate system. Then a polyhedral terrain is the image of a piecewise-linear function in x and y variables. The polyhedral terrain is a generalization
Polyhedral_terrain
Concept in mathematics
unit interval. But unlike the sine and cosine functions, which are continuous, Walsh functions are piecewise constant. They take the values −1 and +1 only
Walsh_function
Higher tax on richer sources
The function which defines the progressive approach to an income tax may be mathematically defined as a piecewise function. In every piece (tax bracket)
Progressive_tax
Assembly of layers of fibrous composite materials
Once part of a laminate, the transformed elasticity is treated as a piecewise function along the z axis (in the thickness direction), hence the integration
Composite_laminate
Functional square root of an exponential
there is no semi-exponential function f(x) that is both (a) analytic and (b) always maps reals to reals. The piecewise solution above achieves goal (b)
Half-exponential_function
Curve whose range contains the unit square
1-dimensionality to curves; all normally encountered curves were piecewise differentiable (that is, have piecewise continuous derivatives), and such curves cannot fill
Space-filling_curve
Non-sinusoidal waveform
waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function. Like a square wave, the triangle wave contains only odd
Triangle_wave
American mathematician (born 1937)
an energy minimization approach allowing optimal computation for piecewise function solutions. Another assessment in 2024 was "a foundational approach
David_Mumford
Theorem in vector calculus
let c0, c1: [0, 1] → U be piecewise smooth loops. If there is a function H: [0, 1] × [0, 1] → U such that [TLH0] H is piecewise smooth, [TLH1] H(t, 0) =
Stokes'_theorem
Method of multivariate interpolation
does not consider the values of neighboring points at all, yielding a piecewise-constant interpolant. The algorithm is very simple to implement and is
Nearest-neighbor interpolation
Nearest-neighbor_interpolation
Statistical distribution
well-modeled by different relations in different regions; on the whole by a piecewise function. For example, if two different power laws fit better in different
Rank–size_distribution
Computational statistics technique
least, closer to piecewise linear). Instead of a single uniform envelope density function, use a piecewise linear density function as your envelope instead
Rejection_sampling
details. Let K and L be two GSCs. A function f : | K | → | L | {\displaystyle f:|K|\to |L|} is called piecewise-linear (PL) if there exist a subdivision
Simplicial_map
Replacing a number with a simpler value
same value). In the general case of a discrete range, they are piecewise constant functions. Typical rounding problems include: The most basic form of rounding
Rounding
Integral criterion for holomorphy
piecewise C1 curve such that γ ( 0 ) = z 0 {\displaystyle \gamma (0)=z_{0}} and γ ( 1 ) = z {\displaystyle \gamma (1)=z} . Then define the function F
Morera's_theorem
Statistical method
Springer. Mumford, D., & Shah, J. (1989). Optimal approximations by piecewise smooth functions and associated variational problems. Communications on pure and
Step_detection
Concept in statistical mathematics
Segmented regression, also known as piecewise regression or broken-stick regression, is a method in regression analysis in which the independent variable
Segmented_regression
literature, with few exceptions, e.g., are for only either linear or piecewise linear (including conventional deadline-based) TUFs because they are easier
Time-utility_function
Model in digital signal processing
impulse response) so that each input impulse results in the correct piecewise linear function in the output. First-order hold is the hypothetical filter or
First-order_hold
Arithmetic operation
0\\a^{\left(^{x-1}a\right)}&0<x\end{cases}}} hence: and so on. However, it is only piecewise differentiable; at integer values of x, the derivative is multiplied by
Tetration
Property of functions which is weaker than continuity
( y , ∞ ] {\displaystyle (y,\infty ]} . Consider the function f , {\displaystyle f,} piecewise defined by: f ( x ) = { − 1 if x < 0 , 1 if x ≥ 0 {\displaystyle
Semi-continuity
test function". Journal of Global Optimization. 25 (2): 169–174. doi:10.1023/A:1021956306041. Bosse, Torsten F.; Bücker, H. Martin (2026). "A piecewise smooth
Griewank_function
Largest and smallest value taken by a function at a given point
derivative test, given sufficient differentiability. For any function that is defined piecewise, one finds a maximum (or minimum) by finding the maximum (or
Maximum_and_minimum
Alternative representations of the RGB color model
{\displaystyle (R,G,B)=(R_{1}+m,G_{1}+m,B_{1}+m)} The polygonal piecewise functions can be somewhat simplified by clever use of minimum and maximum values
HSL_and_HSV
Number whose square is a given number
iterative method that is close to the square root, for which a polynomial or piecewise-linear approximation can be used. The time complexity for computing a
Square_root
Operation in calculus
Although all bounded piecewise continuous functions are Riemann-integrable on a bounded interval, subsequently more general functions were considered—particularly
Integral
Mathematical theorem about functions
x ∈ R n {\displaystyle x\in \mathbb {R} ^{n}} . Piecewise smooth; one dimension If the function is absolutely integrable in one dimension (i.e. f
Fourier_inversion_theorem
Algorithm for measuring similarity between temporal sequences
and monotonicity of time warp functions may be obtained for instance by integrating a time-varying radial basis function, thus being a one-dimensional
Dynamic_time_warping
Representation of mathematical space
called a triangulable space. Triangulations can also be used to define a piecewise linear structure for a space, if one exists. Triangulation has various
Triangulation_(topology)
Arctangent function with two arguments
π]. In terms of the standard arctangent function, whose image is (−12π,12π), atan2 can be expressed piecewise: atan2(y,x)={arctan(yx)if x>0
Atan2
First known wavelet basis
k<2^{n}.} These functions sn,k are continuous, piecewise linear, supported by the interval In,k that also supports ψn,k. The function sn,k is equal to 1
Haar_wavelet
Smooth approximation to the maximum function
RealSoftMax or multivariable softplus) function is a smooth maximum – a smooth approximation to the maximum function, mainly used by machine learning algorithms
LogSumExp
Material whose resistance changes when a force is applied
current I {\displaystyle I} with V F S R {\displaystyle V_{FSR}} are piecewise functions just as the Simmons equations are: When V b u l k ≈ 0 {\displaystyle
Force-sensing_resistor
Method of mathematical integration
functions, mainly piecewise continuous functions, including elementary functions, for example polynomials. However, the graphs of other functions, for example
Lebesgue_integral
Topological manifold with a piecewise smooth structure
smooth manifolds and smooth functions between them) and in PL (the category of piecewise linear manifolds and piecewise linear maps between them), allowing
Piecewise-smooth_manifold
Geometric model of the planar projection of the physical universe
positively oriented, piecewise smooth, simple closed curve in a plane, and let D be the region bounded by C. If L and M are functions of (x, y) defined on
Euclidean_plane
Fundamental theorem in probability theory and statistics
realizations of the sum of n independent identical discrete variables, the piecewise-linear curve that joins the centers of the upper faces of the rectangles
Central_limit_theorem
Curve simplification algorithm
Assuming the input is a one-based array: # source: https://karthaus.nl/rdp/ function DouglasPeucker(PointList[], epsilon) # Find the point with the maximum
Ramer–Douglas–Peucker algorithm
Ramer–Douglas–Peucker_algorithm
electronic oscillators. Consider feedback around a discontinuous (but piecewise continuous) nonlinearity (e.g., an amplifier with saturation, or an element
Describing_function
Finite volume method in partial differential equations
} The function ϕ ( r i ) {\displaystyle \phi \left(r_{i}\right)} is a limiter function that limits the slope of the piecewise approximations
MUSCL_scheme
Mathematical function conceived as a crude model
they may also take the form of other nonlinear functions, piecewise linear functions, or step functions. They are also often monotonically increasing,
Artificial_neuron
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PIECEWISE FUNCTION
PIECEWISE FUNCTION
PIECEWISE FUNCTION
PIECEWISE FUNCTION
PIECEWISE FUNCTION
PIECEWISE FUNCTION
PIECEWISE FUNCTION
PIECEWISE FUNCTION
PIECEWISE FUNCTION
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