# Constant function

Constant function y=4

In mathematics, a constant function is a function whose (output) value is the same for every input value.[1][2][3] For example, the function ${\displaystyle y(x)=4}$ is a constant function because the value of  ${\displaystyle y(x)}$  is 4 regardless of the input value ${\displaystyle x}$ (see image).

## Basic properties

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

Example: The function  ${\displaystyle y(x)=2}$  or just  ${\displaystyle y=2}$  is the specific constant function where the output value is  ${\displaystyle c=2}$. 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,.... 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 euro.

The graph of the constant function ${\displaystyle y=c}$ is a horizontal line in the plane that passes through the point ${\displaystyle (0,c)}$.[4]

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 ${\displaystyle f(x)=c\,,\,\,c\neq 0}$ . This function has no intersection point with the x-axis, that is, it has no root (zero). On the other hand, the polynomial  ${\displaystyle f(x)=0}$  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.[5]

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.[6] This is often written:  ${\displaystyle (c)'=0}$ . The converse is also true. Namely, if y'(x)=0 for all real numbers x, then y(x) is a constant function.[7]

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

## 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.

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

## References

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