# Constant function

Short description: Type of mathematical function

In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function y(x) = 4 is a constant function because the value of y(x) is 4 regardless of the input value x (see image).

## Basic properties

As a real-valued function of a real-valued argument, a constant function has the general form y(x) = c or just y = c.

Example: The function y(x) = 2 or just y = 2 is the specific constant function where the output value is c = 2. The domain of this function is the set of all real numbers R. The codomain of this function is just {2}. 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) = 2, y(−2.7) = 2, y(π) = 2, and so on. No matter what value of x is input, the output is "2".
Real-world example: A store where every item is sold for the price of 1 dollar.

The graph of the constant function y = c is a horizontal line in the plane that passes through the point (0, c).

In the context of a polynomial in one variable x, the non-zero constant function 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, that is, 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.

A constant function is an even function, i.e. the graph of a constant function is symmetric with respect to the y-axis.

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. This is often written: $\displaystyle{ (x \mapsto c)' = 0 }$. The converse is also true. Namely, if y′(x) = 0 for all real numbers x, then y is a constant function.

Example: Given the constant function $\displaystyle{ y(x) = -\sqrt{2} }$. The derivative of y is the identically zero function $\displaystyle{ y'(x) = \left(x \mapsto -\sqrt{2}\right)' = 0 }$.

## 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.
• For any non-empty Y, every set X is isomorphic to the set of constant functions in $\displaystyle{ Y \to X }$. For any Y and each element x in X, there is a unique function $\displaystyle{ \tilde{x}: Y \to X }$ such that $\displaystyle{ \tilde{x}(y) = x }$ for all $\displaystyle{ y \in Y }$. Conversely, if a function $\displaystyle{ f: Y \to X }$ satisfies $\displaystyle{ f(y) = f\left(y'\right) }$ for all $\displaystyle{ y, y' \in Y }$, $\displaystyle{ f }$ is by definition a constant function.
• Every set $\displaystyle{ X }$ is canonically isomorphic to the function set $\displaystyle{ X^1 }$, or hom set $\displaystyle{ \operatorname{hom}(1,X) }$ 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, $\displaystyle{ \operatorname{hom}(X \times Y, Z) \cong \operatorname{hom}(X(\operatorname{hom}(Y, Z)) }$) 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 $\displaystyle{ \lambda: 1 \times X \cong X \cong X \times 1: \rho }$ natural in X, the left and right unitors are the projections $\displaystyle{ p_1 }$ and $\displaystyle{ p_2 }$ the ordered pairs $\displaystyle{ (*, x) }$ and $\displaystyle{ (x, *) }$ respectively to the element $\displaystyle{ x }$, where $\displaystyle{ * }$ is the unique point in the one-point set.