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Work by Gottlob Frege
"Function and Concept" (German: "Funktion und Begriff") is a lecture delivered by Gottlob Frege in Jena at a meeting of the Jenaische Gesellschaft für
Function_and_Concept
About mathematical functions
The mathematical concept of a function dates from the 17th century in connection with the development of calculus; for example, the slope d y / d x {\displaystyle
History of the function concept
History_of_the_function_concept
Association of one output to each input
planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th
Function_(mathematics)
Mathematical function with no sudden changes
concepts of calculus and mathematical analysis, where arguments and values of functions are real numbers and complex numbers. The concept has been generalized
Continuous_function
Point to which functions converge in analysis
mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input which
Limit_of_a_function
Order-preserving mathematical function
a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in
Monotonic_function
Fundamental unit of cognition
A concept is a fundamental unit of cognition that classifies entities and encodes shared features. Concepts make it possible to form and combine ideas
Concept
Linear map or polynomial function of degree one
polynomial function of degree zero (a constant polynomial) or one (a linear polynomial). For distinguishing such a linear function from the other concept, the
Linear_function
Branch of mathematics
corresponding concept in mechanics is the principle of least/stationary action. Functionals are often expressed as definite integrals involving functions and their
Calculus
Transforming a function in such a way that it only takes a single argument
In mathematics and computer science, currying (named after Haskell Curry) is the technique of translating a function that takes multiple arguments into
Currying
Relationship of a signal transducer
susceptibility, impulse response or impedance; see also transfer function. The concept of a Green's function or fundamental solution of an ordinary differential equation
Linear_response_function
Functions such that f(–x) equals f(x) or –f(x)
decomposed as the sum of an even function and an odd function. The concept of even and odd functions appears to date back to the early 18th century, with
Even_and_odd_functions
German philosopher, logician, and mathematician (1848–1925)
Enquiry into the Concept of Number, 2nd ed. Blackwell. 1891. "Funktion und Begriff." Translation: "Function and Concept" in Geach and Black (1980). 1892a
Gottlob_Frege
Category of cloud computing services
Function as a service is a "platform-level cloud capability" that enables its users "to build and manage microservices applications with low initial investment
Function_as_a_service
Topics referred to by the same term
computer keyboards Function model, a structured representation of processes in a system Function object or functor or functionoid, a concept of object-oriented
Function
written De Branges space) is a concept in functional analysis and is constructed from a de Branges function. The concept is named after Louis de Branges
De_Branges_space
Philosophical problem about Frege's distinction between concept and object
predication, higher-order reference and type theory. In Frege's mature work, concepts are treated as special kinds of functions from objects to truth-values
Concept_horse_paradox
Construct related to weighted sums and averages
function is a weighted sum or weighted average. Weight functions occur frequently in statistics and analysis, and are closely related to the concept of
Weight_function
Function that preserves distinctness
In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct
Injective_function
In probability, a theory
du} is the complementary cumulative distribution function (also called survival function). The concept is named after John P. Mills. The Mills ratio is
Mills_ratio
Whole number
of a zero object, often denoted 0, and the related concept of zero morphisms, which generalize the zero function. Ancient Egyptian numerals were of base
0
Mathematical relation consisting of a multi-variable function equal to zero
circle defines y as an implicit function of x, y = 1 − x 2 {\displaystyle y={\sqrt {1-x^{2}}}} , assuming −1 ≤ x ≤ 1 and y is restricted to nonnegative
Implicit_function
Method of mathematical integration
mathematician Henri Lebesgue, is one way to make this concept rigorous and to extend it to more general functions. The Lebesgue integral is more general than the
Lebesgue_integral
Relationship between derivatives and integrals
links the concept of differentiating a function (calculating its slopes, or rate of change at every point on its domain) with the concept of integrating
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Function with a repeating pattern
function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which are used to describe waves and other
Periodic_function
On converting relations to functions of several real variables
{\displaystyle y} algebraically, and the implicit function theorem gives analytic conditions under which there exists a function f {\displaystyle f} whose graph
Implicit_function_theorem
Course designed to prepare students for calculus
fundamental concepts of variables and functions. His innovation is noted for its use of exponentiation to introduce the transcendental functions. The general
Precalculus
Cost-minimizing level of an input in economics
so output is one argument of the function. This concept typically arises in a long run context in which both labor and capital usage are choosable by the
Conditional_factor_demands
Theoretical framework
studies are relevant to various stages of concept formation. Semantics is fundamentally a study of concepts, the meaning that thinking beings give to
Conceptual_model
Programming construct
related to the functional programming concept). A typical use of a function object is in writing callback functions. A callback in procedural languages
Function_object
Extension to C++ templates
evaluated at compile time. A concept may be associated with a template (class template, function template, member function of a class template, variable
Concepts_(C++)
Operation in calculus
negative. Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function; in this case, they are also called
Integral
Concept in economics and decision theory
maximize, i.e., an objective function. This kind of utility bears a closer resemblance to the original utilitarian concept, developed by moral philosophers
Utility
Concept in the analysis of dynamical systems
theory. A similar concept appears in the theory of general state-space Markov chains usually under the name Foster–Lyapunov functions. For certain classes
Lyapunov_function
Mathematical relation assigning a probability event to a cost
mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or
Loss_function
Topics referred to by the same term
Distribution function may refer to Cumulative distribution function, a basic concept of probability theory Distribution function (physics), a function giving
Distribution_function
Special mathematical function defined as sin(x)/x
entire function. The normalized sinc function is the Fourier transform of the rectangular function with no scaling. It is used in the concept of reconstructing
Sinc_function
Mathematical approximation of a function
function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series
Taylor_series
Notion in calculus
calculus, the differential represents the principal part of the change in a function y = f ( x ) {\displaystyle y=f(x)} with respect to changes in the independent
Differential_of_a_function
Limit of a function One-sided limit Limit of a sequence Indeterminate form Orders of approximation (ε, δ)-definition of limit Continuous function Derivative
List_of_calculus_topics
Design philosophy of 19th–20th centuries
Sullivan also credited his friend and mentor, John H. Edelmann, who theorized the concept of "suppressed function" with inspiration for this maxim. The
Form_follows_function
Largest and smallest value taken by a function at a given point
maximum. In statistics, the corresponding concept is the sample maximum and minimum. A real-valued function f defined on a domain X has a global (or absolute)
Maximum_and_minimum
Matrix of partial derivatives of a vector-valued function
the number of components of function values, then its determinant is called the Jacobian determinant. Both the matrix and (if applicable) the determinant
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Theorem in mathematics
mathematical analysis, the inverse function theorem gives sufficient conditions for a function to have an inverse function. The essential idea is that if
Inverse_function_theorem
Mathematical notion of infinitesimal difference
and the derivatives of functions. The term is used in various branches of mathematics such as calculus, differential geometry, algebraic geometry and
Differential_(mathematics)
Mapping arbitrary data to fixed-size values
differently. The hash function differs from these concepts mainly in terms of data integrity. Hash tables may use non-cryptographic hash functions, while cryptographic
Hash_function
In functional programming
is a function that takes two arguments. Partial application is sometimes incorrectly called currying, which is a related, but distinct concept. Intuitively
Partial_application
Mathematical rule for evaluating limits
tutor, the Swiss mathematician Johann Bernoulli. For two functions f {\displaystyle f} and g {\displaystyle g} , under most circumstances the limit of
L'Hôpital's_rule
Concept in programming language design
being passed as an argument, returned from a function, and assigned to a variable. The concept of first- and second-class objects was introduced by Christopher
First-class_citizen
Distinction in the philosophy of language
mathematical function) led to his views on a theory of reference. Frege developed his original theory of meaning in early works like Begriffsschrift (concept paper)
Sense_and_reference
Concept in design processes
Form, Fit, and Function (also F3 or FFF) is a concept used in various industries, including manufacturing, engineering, and architecture, to describe aspects
Form,_fit_and_function
Integrals not expressible in closed-form from elementary functions
antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an elementary function. A theorem by Liouville in
Nonelementary_integral
function. The concept was first defined by Iimura. Some variants of it were later defined by Yang, Chen and Deng, Herings, van-der-Laan, Talman and Yang
Direction-preserving_function
Study of rates of change
are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen input equals
Differential_calculus
Distance from a point to the boundary of a set
outside). The concept also sometimes goes by the name oriented distance function/field. Let Ω be a subset of a metric space X with metric d, and ∂ Ω {\displaystyle
Signed_distance_function
Term in educational psychology
Concept learning, also known as category learning, concept attainment, and concept formation, is defined by Bruner, Goodnow, & Austin (1956) as "the search
Concept_learning
g(x, y) instead of x.g(y). Friend functions have the same implications on encapsulation as methods. A similar concept is that of friend class. This approach
Friend_function
Testing software functionality
naming, functional testing is not testing the code of a single function. The concept of incorporating testing earlier in the delivery cycle is not restricted
Functional_testing
Reason for a change under natural selection; in physiology, what a system does
sensation or locomotion in an animal. This concept of function as opposed to form (respectively Aristotle's ergon and morphê) was central in biological explanations
Function_(biology)
Topological space that locally resembles Euclidean space
naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to
Manifold
Public-key cryptographic pseudorandom function
necessarily seem random. The concept of a VRF was introduced by Micali, Rabin, and Vadhan in 1999. Since then, verifiable random functions have found widespread
Verifiable_random_function
Historical mathematical concept; form of derivative
time). Newton introduced the concept in 1665 and detailed them in his mathematical treatise, Method of Fluxions. Fluxions and fluents made up Newton's early
Fluxion
Object that creates other objects
a factory is an object for creating other objects; formally, it is a function or method that returns objects of a varying prototype or class from some
Factory (object-oriented programming)
Factory_(object-oriented_programming)
Value approached by a mathematical object
that a function (or sequence) approaches as the argument (or index) approaches some value. Limits of functions are essential to calculus and mathematical
Limit_(mathematics)
Integration method to calculate volume
the function and the axis of rotation. This works only if the axis of rotation is horizontal (example: y = 3 or some other constant). If the function to
Disc_integration
Method of solution to differential equations
equations and diffusion equations. In quantum mechanics, Green's function of the Hamiltonian is a key concept with important links to the concept of density
Green's_function
Subset of a function's codomain
the range of a function may refer to either of two closely related concepts: the codomain of the function, or the image of the function. In some cases
Range_of_a_function
Mathematical concept in measure theory
mathematical analysis and measure theory, an approximately continuous function is a concept that generalizes the notion of continuous functions by replacing the
Approximately continuous function
Approximately_continuous_function
Generalization of the concept of directional derivative
generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, it is defined for functions between locally convex
Gateaux_derivative
One-to-one correspondence
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the
Bijection
Album that tells a self-contained story
and we needed to build some filler around it. Each tune had a function." The Beatles' John Lennon commented: "Sgt. Pepper is called the first concept
Concept_album
Matrix of second derivatives
vector-valued functionPages displaying short descriptions of redirect targets Hessian equation Binmore, Ken; Davies, Joan (2007). Calculus Concepts and Methods
Hessian_matrix
Mathematical concept
inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if and only if f
Inverse_function
Instantaneous rate of change (mathematics)
functions and many other functions can be differentiated using a concept known as the weak derivative. The idea is to embed the continuous functions in
Derivative
Multivariate derivative (mathematics)
learning, and artificial intelligence, where it is used to minimize a function by gradient descent. In coordinate-free terms, the gradient of a function f (
Gradient
Expression which is not assigned an interpretation
complex geometric concepts. Contrast also the term undefined behavior in computer science, in which the term indicates that a function may produce or return
Undefined_(mathematics)
Function from the limited hyperreal to the real numbers
historical concept of adequality introduced by Pierre de Fermat, as well as Leibniz's transcendental law of homogeneity. The standard part function was first
Standard_part_function
Operation on mathematical functions
two functions, f {\displaystyle f} and g {\displaystyle g} , and returns a new function f ∘ g {\displaystyle f\circ g} . When the composite function f ∘
Function_composition
Formula for the derivative of an inverse function
calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the
Inverse_function_rule
Function that returns its argument unchanged
the concept of an identity morphism in category theory, where the endomorphisms of M {\displaystyle M} need not be functions. The identity function is
Identity_function
Hypothesis in cognitive neuroscience
associated with these functions. The concept of neuronal recycling resolves this paradox by suggesting that novel functions actually utilize and 'recycle' existing
Neuronal_recycling_hypothesis
Process by which a quantum system takes on a definitive state
vector" (or "state reduction" for short) and "wave function collapse" are used to describe the same concept. A quantum state is a mathematical description
Wave_function_collapse
Time-varying quantity or variable
his early calculus to describe his form of a function. The concept was introduced by Newton in 1665 and detailed in his mathematical treatise, Method
Fluent_(mathematics)
Mathematical operation in calculus
(1998). Calculus Concepts And Contexts. Brooks/Cole Publishing Company. ISBN 0-534-34330-9. Apostol, Tom M. (1974). "13. Implicit functions and extremum problems"
Implicit_differentiation
Interrelated entities that form a whole
surrounded and influenced by its environment, is described by its boundaries, structure and purpose and is expressed in its functioning. Systems are
System
Graphical depicture of loss
quality, and helped fuel the continuous improvement movement. The concept of Taguchi's quality loss function was in contrast with the American concept of quality
Taguchi_loss_function
Linear combination of indicator functions of real intervals
mathematical concept behind some test signals, such as those used to determine the step response of a dynamical system. The rectangular function, the normalized
Step_function
Family of solutions to related differential equations
Bessel functions are a class of special functions that commonly appear in problems involving wave motion, heat conduction, and other physical phenomena
Bessel_function
Scientific principles enabling the use of the calculus of variations
solved using variational calculus, and in this case, the variational principle is the following: The solution is a function that minimizes the gravitational
Variational_principle
Computational learning theory
theory in mathematics, a concept over a domain X is a total Boolean function over X. A concept class is a class of concepts. Concept classes are a subject
Concept_class
Process of repeating items in a self-similar way
most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While
Recursion
Used to define marginal product and to distinguish allocative efficiency
production function gives the technological relation between quantities of physical inputs and quantities of output of goods. The production function is one
Production_function
Formula in calculus
derivative of the composition of two differentiable functions z and y in terms of the derivatives of z and y. More precisely, if h = z ∘ y {\displaystyle h=z\circ
Chain_rule
Quantum state of the entire universe
mechanics, and finds applications in quantum cosmology. It evolves deterministically according to a wave equation. The concept of universal wave function was
Universal_wave_function
Mathematical method in calculus
that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. It is frequently used
Integration_by_parts
Indefinite integral
function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function
Antiderivative
Method for calculating the volume of a solid of revolution
) d x {\displaystyle 2\pi \int _{a}^{b}xf(x)\,dx} If the function is of the y coordinate and the axis of rotation is the x-axis then the formula becomes:
Shell_integration
Mapping which preserves all topological properties of a given space
trefoil knot and a circle. Homotopy and isotopy are precise definitions for the informal concept of continuous deformation. A function f : X → Y {\displaystyle
Homeomorphism
floor and ceiling functions, the Dirichlet function, the sign function and the Heaviside step function (except possibly at 0). Integer-valued functions defined
Integer-valued_function
Derivative of a function with multiple variables
Partial derivatives are used in vector calculus and differential geometry. The partial derivative of a function f ( x , y , … ) {\displaystyle f(x,y,\dots
Partial_derivative
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