# Constant function

mathematical function whose (output) value is the same for every input value

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

Formally, a constant function f(x):RR has the form  ${\displaystyle f(x)=c}$ . Usually we write ${\displaystyle y(x)=c}$   or just  ${\displaystyle y=c}$ .

• The function y=c has 2 variables x and у and 1 constant c. (In this form of the function, we do not see x, but it is there.)
• The constant c is a real number. Before working with a linear function, we replace c with an actual number.
• The domain or input of y=c is R. So any real number x can be input. However, the output is always the value c.
• The range of y=c is also R. However, because the output is always the value of c, the codomain is just c.

Example: The function  ${\displaystyle y(x)=4}$   or just  ${\displaystyle y=4}$   is the specific constant function where the output value is  ${\displaystyle c=4}$ . The domain is all real numbers ℝ. The codomain is just {4}. Namely, y(0)=4, y(−2.7)=4, y(π)=4,.... No matter what value of x is input, the output is "4".

• 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]
• If c≠0, the constant function y=c is a polynomial in one variable x of degree zero.
• The y-intercept of this function is the point (0,c).
• This function has no x-intercept. That is, it has no root or zero. It never crosses the x-axis.
• If c=0, then we have y=0. This is the zero polynomial or the identically zero function. Every real number x is a root. The graph of y=0 is the x-axis in the plane.[5]
• A constant function is an even function so the y-axis is an axis of symmetry for every constant function.

## Derivative of a constant function

In the context where it is defined, the derivative of a function measures the rate of change of function (output) values with respect to change in input values. A constant function does not change, so its derivative is 0.[6] This is often written:  ${\displaystyle (c)'=0}$  .

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

The converse (opposite) is also true. That is, if the derivative of a function is zero everywhere, then the function is a constant function.[7]

Mathematically we write these two statements:

${\displaystyle y(x)=c\,\,\,\Leftrightarrow \,\,\,y'(x)=0\,,\,\,\forall x\in \mathbb {R} }$

## Generalization

A function f : AB is a constant function if f(a) = f(b) for every a and b in A.[8]

## Examples

Real-world example: A store where every item is sold for 1 euro. The domain of this function is items in the store. The codomain is 1 euro.

Example: Let f : AB where A={X,Y,Z,W} and B={1,2,3} and f(a)=3 for every aA. Then f is a constant function.

Example: z(x,y)=2 is the constant function from A=ℝ² to B=ℝ where every point (x,y)∈ℝ² is mapped to the value z=2. The graph of this constant function is the horizontal plane (parallel to the x0y plane) in 3-dimensional space that passes through the point (0,0,2).

Example: The polar function ρ(φ)=2.5 is the constant function that maps every angle φ to the radius ρ=2.5. The graph of this function is the circle of radius 2.5 in the plane.

 Generalized constant function. Constant function z(x,y)=2 Constant polar function ρ(φ)=2.5

## Other properties

There are other properties of constant functions. See Constant function on English Wikipedia

## References

1. Tanton, James (2005). Encyclopedia of Mathematics. Facts on File, New York. p. 94. ISBN 0-8160-5124-0. (in English)
2. C.Clapham, J.Nicholson (2009). "Oxford Concise Dictionary of Mathematics, Constant Function" (PDF). Addison-Wesley. p. 175. Retrieved 1 January 2014.
3. Weisstein, Eric (1999). CRC Concise Encyclopedia of Mathematics. CRC Press, London. p. 313. ISBN 0-8493-9640-9. (in English)
4. Dawkins, Paul (2007). "College Algebra". Lamar University. p. 224. Retrieved 1 January 2014.
5. Carter, John A.; Cuevas, Gilbert J.; Holliday, Berchie; Marks, Daniel; McClure, Melissa S.publisher=Glencoe/McGraw-Hill School Pub Co (2005). Advanced Mathematical Concepts - Pre-calculus with Applications, Student Edition 1. p. 22. ISBN 978-0078682278. (in English)
6. Dawkins, Paul (2007). "Derivative Proofs". Lamar University. Retrieved 1 January 2014.
7. "Zero Derivative implies Constant Function". Retrieved 1 January 2014.
8. "Constant Function". Retrieved 1 January 2014.