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Constant function

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From Wikipedia, the free encyclopedia

Type of mathematical function

Not to be confused withfunction constant.

Function
x ↦f (x)
History of the function concept
Types bydomain andcodomain
`X` →𝔹 𝔹 →`X` 𝔹ⁿ →`X` `X` →ℤ ℤ →`X` `X` →ℝ ℝ →`X` ℝⁿ →`X` `X` →ℂ ℂ →`X` ℂⁿ →`X`
Classes/properties
Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions
Restriction Composition λ Inverse
Generalizations
Relation (Binary relation) Set-valued Multivalued Partial Implicit Space Higher-order Morphism Functor
List of specific functions
v t e

Inmathematics, aconstant function is afunction whose (output) value is the same for every input value.

Basic properties

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An example of a constant function isy(x) = 4, because the value ofy(x) is 4 regardless of the input valuex.

As a real-valued function of a real-valued argument, a constant function has the general formy(x) =c or justy =c. For example, the functiony(x) = 4 is the specific constant function where the output value isc = 4. Thedomain of this function is the set of allreal numbers. Theimage of this function is thesingleton set{4}. The independent variablex does not appear on the right side of the function expression and so its value is "vacuously substituted"; namelyy(0) = 4,y(−2.7) = 4,y(π) = 4, and so on. No matter what value ofx is input, the output is4.^[1]

The graph of the constant functiony =c is ahorizontal line in theplane that passes through the point(0,c).^[2] In the context of apolynomial in one variablex, the constant function is callednon-zero constant function because it is a polynomial of degree 0, and its general form isf(x) =c, wherec is nonzero. This function has no intersection point with thex-axis, meaning it has noroot (zero). On the other hand, the polynomialf(x) = 0 is theidentically zero function. It is the (trivial) constant function and everyx is a root. Its graph is thex-axis in the plane.^[3] Its graph is symmetric with respect to they-axis, and therefore a constant function is aneven function.^[4]

In the context where it is defined, thederivative of a function is a measure of the rate of change of function values with respect to change in input values. Because a constant function does not change, its derivative is 0.^[5] This is often written: $(x\mapsto c)'=0$ . The converse is also true. Namely, ify′(x) = 0 for all real numbersx, theny is a constant function.^[6] For example, given the constant function $y(x)=-{\sqrt {2}}$ . The derivative ofy is the identically zero function $y'(x)=\left(x\mapsto -{\sqrt {2}}\right)'=0$ .

Other properties

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For functions betweenpreordered sets, constant functions are bothorder-preserving andorder-reversing; conversely, iff is both order-preserving and order-reversing, and if thedomain off is alattice, thenf must be constant.

Every constant function whosedomain andcodomain are the same setX is aleft zero of thefull transformation monoid onX, which implies that it is alsoidempotent.
It has zeroslope orgradient.
Every constant function betweentopological spaces iscontinuous.
A constant function factors through theone-point set, theterminal object in thecategory of sets. This observation is instrumental forF. William Lawvere's axiomatization of set theory, theElementary Theory of the Category of Sets (ETCS).^[7]
For any non-emptyX, every setY isisomorphic to the set of constant functions in $X\to Y$ . For anyX and each elementy inY, there is a unique function ${\tilde {y}}:X\to Y$ such that ${\tilde {y}}(x)=y$ for all $x\in X$ . Conversely, if a function $f:X\to Y$ satisfies $f(x)=f(x')$ for all $x,x'\in X$ , $f {\displaystyle f}$ is by definition a constant function.
- As a corollary, the one-point set is agenerator in the category of sets.
- Every set $X {\displaystyle X}$ is canonically isomorphic to the function set $X^{1}$ , orhom set $\operatorname {hom} (1,X)$ in the category of sets, where 1 is the one-point set. Because of this, and the adjunction between Cartesian products and hom in the category of sets (so there is a canonical isomorphism between functions of two variables and functions of one variable valued in functions of another (single) variable, $\operatorname {hom} (X\times Y,Z)\cong \operatorname {hom} (X(\operatorname {hom} (Y,Z))$ ) the category of sets is aclosed monoidal category with theCartesian product of sets as tensor product and the one-point set as tensor unit. In the isomorphisms $\lambda :1\times X\cong X\cong X\times 1:\rho$ natural inX, the left and right unitors are the projections $p_{1}$ and $p_{2}$ theordered pairs $(*,x)$ and $(x,*)$ respectively to the element $x {\displaystyle x}$ , where $*$ is the uniquepoint in the one-point set.

A function on aconnected set islocally constant if and only if it is constant.

References

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^Tanton, James (2005).Encyclopedia of Mathematics. Facts on File, New York. p. 94.ISBN 0-8160-5124-0.
^Dawkins, Paul (2007)."College Algebra". Lamar University. p. 224. RetrievedJanuary 12, 2014.
^Carter, John A.; Cuevas, Gilbert J.; Holliday, Berchie; Marks, Daniel; McClure, Melissa S. (2005). "1".Advanced Mathematical Concepts - Pre-calculus with Applications, Student Edition (1 ed.). Glencoe/McGraw-Hill School Pub Co. p. 22.ISBN 978-0078682278.
^Young, Cynthia Y. (2021).Precalculus (3rd ed.). John Wiley & Sons. p. 122.ISBN 978-1-119-58294-6.
^Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007).Calculus (9th ed.).Pearson Prentice Hall. p. 107.ISBN 978-0131469686.
^"Zero Derivative implies Constant Function". RetrievedJanuary 12, 2014.
^Leinster, Tom (27 Jun 2011). "An informal introduction to topos theory".arXiv:1012.5647 [math.CT].

Herrlich, Horst and Strecker, George E.,Category Theory, Heldermann Verlag (2007).

External links

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Wikimedia Commons has media related toConstant functions.

v t e Polynomials andpolynomial functions andpolynomial equations
Bydegree	Zero polynomial (degree undefined or −1 or −∞) Constant function (0) Linear function (1) Linear equation Quadratic function (2) Quadratic equation Cubic function (3) Cubic equation Quartic function (4) Quartic equation Quintic function (5) Sextic equation (6) Septic equation (7)
By properties	Univariate Bivariate Multivariate Monomial Binomial Trinomial Irreducible Square-free Homogeneous Quasi-homogeneous
Tools and algorithms	Factorization Greatest common divisor Division Horner's method of evaluation Polynomial identity testing Resultant Discriminant Gröbner basis

v t e Function
History List of specific functions
Types by domain and codomain	X → 𝔹 𝔹 → X 𝔹ⁿ → 𝔹 X → ℤ ℤ → X X → ℝ ℝ → X ℝⁿ → X X → ℂ ℂ → X ℂⁿ → X
Classes/properties	Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions	Restriction Composition λ Inverse
Generalizations	Relation (Binary relation) Set-valued Multivalued Partial Implicit Space Higher-order Morphism Functor
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