In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, a
constant function is a
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
whose values do not vary and thus are
constantIn mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition to variable, which is a symbol that stands for a value that may vary....
. For example, if we have the function
f(
x) = 4, then
f is constant since
f maps any value to 4. More formally, a function
f :
A →
B is a constant function if
f(
x) =
f(
y) for all
x and
y in
A.
Every
empty functionIn mathematics, an empty function is a function whose domain is the empty set. For each set A, there is exactly one such empty functionThe graph of an empty function is a subset of the Cartesian product ∅×A. Since the product is empty the only such subset is the empty set ∅...
is constant,
vacuouslyA vacuous truth is a truth that is devoid of content because it asserts something about all members of a class that is empty or because it says "If A then B" when in fact A is false...
, since there are no
x and
y in
A for which
f(
x) and
f(
y) are different when
A is the empty set.
In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, a
constant function is a
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
whose values do not vary and thus are
constantIn mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition to variable, which is a symbol that stands for a value that may vary....
. For example, if we have the function
f(
x) = 4, then
f is constant since
f maps any value to 4. More formally, a function
f :
A →
B is a constant function if
f(
x) =
f(
y) for all
x and
y in
A.
Every
empty functionIn mathematics, an empty function is a function whose domain is the empty set. For each set A, there is exactly one such empty functionThe graph of an empty function is a subset of the Cartesian product ∅×A. Since the product is empty the only such subset is the empty set ∅...
is constant,
vacuouslyA vacuous truth is a truth that is devoid of content because it asserts something about all members of a class that is empty or because it says "If A then B" when in fact A is false...
, since there are no
x and
y in
A for which
f(
x) and
f(
y) are different when
A is the empty set. Some find it more convenient, however, to define constant function so as to exclude empty functions.
In the context of
polynomial functionIn mathematics, a polynomial is a finite length expression constructed from variables and constants, by using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents...
s, a non-zero constant function is called a polynomial of degree zero.
Properties
Constant functions can be characterized with respect to
function compositionIn mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...
in two ways.
The following are equivalent:
- f : A → B is a constant function.
- For all functions g, h : C → A, f o g = f o h, (where "o" denotes function composition
In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...
).
- The composition of f with any other function is also a constant function.
The first characterization of constant functions given above, is taken as the motivating and defining property for the more general notion of
constant morphism in
category theoryIn mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows....
.
In contexts where it is defined, the
derivativeIn calculus the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point; for example, the derivative of the position of a vehicle with respect to time is the instantaneous velocity...
of a function measures how that function varies with respect to the variation of some argument. It follows that, since a constant function does not vary, its derivative, where defined, will be zero. Thus for example:
- If f is a real-valued
In mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...
function of a real variableA variable is a symbol that stands for a value that may vary; the term usually occurs in opposition to constant, which is a symbol for a non-varying value, i.e. completely fixed or fixed in the context of use...
, defined on some intervalInterval may refer to:* Interval , a range of numbers * Interval measurements or interval variables in statistics is a level of measurement* Interval , the relationship between two notes...
, then f is constant if and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements. In that it is biconditional, the connective can be likened to the standard material conditional combined with its reverse ; hence the name...
the derivativeIn calculus the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point; for example, the derivative of the position of a vehicle with respect to time is the instantaneous velocity...
of f is everywhere zero.
For functions between
preordered setsIn mathematics, especially in order theory, preorders are binary relations that satisfy certain conditions. For example, all partial orders and equivalence relations are preorders. The name quasiorder is also common for preorders. Other names are pre-order, quasi-order, and quasi order...
, constant functions are both order-preserving and order-reversing; conversely, if
f is both order-preserving and order-reversing, and if the
domainIn mathematics, the domain of a given functionis the set of "input" values for which the function is defined. For instance, the domain of cosine would be all real numbers, while the domain of the square root would be only numbers greater than or equal to 0...
of
f is a
latticeIn mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
, then
f must be constant.
Other properties of constant functions include:
- Every constant function whose domain
In mathematics, the domain of a given functionis the set of "input" values for which the function is defined. For instance, the domain of cosine would be all real numbers, while the domain of the square root would be only numbers greater than or equal to 0...
and codomainIn mathematics, the codomain, or target set, of a function is the set Y into which all of the output of the function is constrained to fall...
are the same is idempotent.
- Every constant function between topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s is continuousIn topology and related areas of mathematics a continuous function is a morphism between topological spaces. Intuitively, this is a function f where a set of points near f always contain the image of a set of points near x...
.
A function on a connected set is locally constant if and only if it is constant.
External links