Constant function

Last updated April 03, 2024

In mathematics, a constant function is a function whose (output) value is the same for every input value.

Basic properties

An example of a constant function is y(x) = 4, because the value of y(x) is 4 regardless of the input value x. Wiki constant function 175 200.png — An example of a constant function is $y (x) = 4$ , because the value of $y (x)$ is 4 regardless of the input value $x$ .

As a real-valued function of a real-valued argument, a constant function has the general form $y (x) = c$ or just $y = c$ . For example, the function $y (x) = 4$ is the specific constant function where the output value is $c = 4$ . The domain of this function is the set of all real numbers. The image of this function is the singleton set ${4}$ . The independent variable x does not appear on the right side of the function expression and so its value is "vacuously substituted"; namely $y (0) = 4$ , $y (-2.7) = 4$ , $y (π) = 4$ , and so on. No matter what value of $x$ is input, the output is $4$ .^[1]

The graph of the constant function $y = c$ is a horizontal line in the plane that passes through the point $(0, c)$ .^[2] In the context of a polynomial in one variable $x$ , the constant function is called non-zero constant function because it is a polynomial of degree 0, and its general form is $f (x) = c$ , where $c$ is nonzero. This function has no intersection point with the $x$ -axis, meaning it has no root (zero). On the other hand, the polynomial $f (x) = 0$ is the identically zero function. It is the (trivial) constant function and every $x$ is a root. Its graph is the $x$ -axis in the plane.^[3] Its graph is symmetric with respect to the $y$ -axis, and therefore a constant function is an even function.^[4]

In the context where it is defined, the derivative 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, if $y'(x) = 0$ for all real numbers $x$ , then $y$ is a constant function.^[6] For example, given the constant function $y(x)=-{\sqrt {2}}$ . The derivative of $y$ is the identically zero function $y'(x)=\left(x\mapsto -{\sqrt {2}}\right)'=0$ .

Other properties

For functions between preordered sets, constant functions are both order-preserving and order-reversing; conversely, if $f$ is both order-preserving and order-reversing, and if the domain of $f$ is a lattice, then $f$ must be constant.

Every constant function whose domain and codomain are the same set $X$ is a left zero of the full transformation monoid on $X$ , which implies that it is also idempotent.
It has zero slope or gradient.
Every constant function between topological spaces is continuous.
A constant function factors through the one-point set, the terminal object in the category of sets. This observation is instrumental for F. William Lawvere's axiomatization of set theory, the Elementary Theory of the Category of Sets (ETCS).^[7]
For any non-empty X, every set Y is isomorphic to the set of constant functions in $X\to Y$ $Constant function$ . For any X and each element y in Y, there is a unique function ${\tilde {y}}:X\to Y$ $Constant function$ such that ${\tilde {y}}(x)=y$ $Constant function$ for all $x\in X$ $Constant function$ . Conversely, if a function $f:X\to Y$ $Constant function$ satisfies $f(x)=f(x')$ $Constant function$ for all $x,x'\in X$ $Constant function$ , $f$ $Constant function$ is by definition a constant function.
- As a corollary, the one-point set is a generator in the category of sets.
- Every set $X$ is canonically isomorphic to the function set $X^{1}$ , or hom 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 a closed monoidal category with the Cartesian 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 in $X$ , the left and right unitors are the projections $p_{1}$ and $p_{2}$ the ordered pairs $(*,x)$ and $(x,*)$ respectively to the element $x$ , where $*$ is the unique point in the one-point set.

A function on a connected set is locally constant if and only if it is constant.

Related Research Articles

In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is not continuous. Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity.

The derivative is a fundamental tool of calculus that quantifies the sensitivity of change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a derivative is called differentiation.

In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve.

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In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in that their internal language is the simply typed lambda calculus. They are generalized by closed monoidal categories, whose internal language, linear type systems, are suitable for both quantum and classical computation.

<span class="mw-page-title-main">Cubic function</span> Polynomial function of degree 3

In mathematics, a cubic function is a function of the form $that is, a polynomial function of degree three. In many texts, the coefficients a, b, c, and d are supposed to be real numbers, and the function is considered as a real function that maps real numbers to real numbers or as a complex function that maps complex numbers to complex numbers. In other cases, the coefficients may be complex numbers, and the function is a complex function that has the set of the complex numbers as its codomain, even when the domain is restricted to the real numbers.$

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates with coefficients in another ring, often a field.

In algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if M and N are left modules over a ring R, then a function $is called an R - module homomorphism or an R - linear map if for any x, y in M and r in R,$

In ring theory, a branch of mathematics, a ring is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, x² = 0 implies x = 0. A commutative algebra over a commutative ring is called a reduced algebra if its underlying ring is reduced.

In mathematics, an even function is a real function such that $for every in its domain. Similarly, an odd function is a function such that for every in its domain.$

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information.

In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures in a one-to-one fashion, often by means of an involution operation: if the dual of $A$ is $B$ , then the dual of $B$ is $A$ . Such involutions sometimes have fixed points, so that the dual of $A$ is $A$ itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry.

In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions P_ε, where ε is a small number, or a vector of small quantities. The infinitesimal conditions are the result of applying the approach of differential calculus to solving a problem with constraints. The name is an analogy to non-rigid structures that deform slightly to accommodate external forces.

In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic structures. The cohomology of groups, Lie algebras, and associative algebras can all be defined in terms of Ext. The name comes from the fact that the first Ext group Ext¹ classifies extensions of one module by another.

In mathematics, a critical point is the argument of a function where the function derivative is zero . The value of the function at a critical point is a critical value.

In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes.

In mathematics, the word constant conveys multiple meanings. As an adjective, it refers to non-variance ; as a noun, it has two different meanings:

References

↑ 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. Retrieved January 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.). p. 122.
↑ 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" . Retrieved January 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

Weisstein, Eric W. "Constant Function". MathWorld .
"Constant function". PlanetMath .

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[1] Tanton, James (2005). Encyclopedia of Mathematics. Facts on File, New York. p. 94. ISBN 0-8160-5124-0.

[2] Dawkins, Paul (2007). "College Algebra". Lamar University. p. 224. Retrieved January 12, 2014.

[3] 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.

[4] Young, Cynthia Y. (2021). Precalculus (3rd ed.). p. 122.

[5] Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007). Calculus (9th ed.). Pearson Prentice Hall. p. 107. ISBN 978-0131469686.

[6] "Zero Derivative implies Constant Function" . Retrieved January 12, 2014.

[7] Leinster, Tom (27 Jun 2011). "An informal introduction to topos theory". arXiv: 1012.5647 [math.CT].

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v t e Polynomials and polynomial functions
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Function
$x \mapsto f (x)$
History of the function concept
Examples of domains and codomains
$X$ → $\mathbb {B}$ , $\mathbb {B}$ → $X$ , $\mathbb {B} ^{n}$ → $X$ $X$ → $\mathbb {Z}$ , $\mathbb {Z}$ → $X$ $X$ → $\mathbb {R}$ , $\mathbb {R}$ → $X$ , $\mathbb {R} ^{n}$ → $X$ $X$ → $\mathbb {C}$ , $\mathbb {C}$ → $X$ , $\mathbb {C} ^{n}$ → $X$
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Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
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Binary relation Partial Multivalued Implicit Space
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