Constant 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,.... 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 $y=c$ 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 $f(x)=c\,,\,\,c\neq 0$ . This function has no intersection point with the x-axis, that is, it has no root (zero). On the other hand, the polynomial  $f(x)=0$ 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:  $(c)'=0$ . The converse is also true. Namely, if y'(x)=0 for all real numbers x, then y(x) is a constant function.

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

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.