# Constant function

Not to be confused with function constant.
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 $y(x) = 4$ 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 $(0,c)$.[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 $f(x) = c \, ,\,\, c \neq 0$ . 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.[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:  $(c)'=0$ . 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   $y(x)=-\sqrt{2}$  . 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.

## References

1. ^ Tanton, James (2005). Encyclopedia of Mathematics. Facts on File, New York. p. 94. ISBN 0-8160-5124-0.
2. ^ C.Clapham, J.Nicholson (2009). "Oxford Concise Dictionary of Mathematics, Constant Function" (in english). Addison-Wesley. p. 175. Retrieved January 2014.
3. ^ Weisstein, Eric (1999). CRC Concise Encyclopedia of Mathematics. CRC Press, London. p. 313. ISBN 0-8493-9640-9.
4. ^ Dawkins, Paul (2007). "College Algebra" (in english). Lamar University. p. 224. Retrieved January 2014.
5. ^ Carter, John A.; Cuevas, Gilbert J.; Holliday, Berchie; Marks, Daniel; McClure, Melissa S.publisher=Glencoe/McGraw-Hill School Pub Co (2005). "1". Advanced Mathematical Concepts - Pre-calculus with Applications, Student Edition (1 ed.). p. 22. ISBN 978-0078682278.
6. ^ Dawkins, Paul (2007). "Derivative Proofs" (in english). Lamar University. Retrieved January 2014.
7. ^ "Zero Derivative implies Constant Function" (in english). Retrieved January 2014.
• Herrlich, Horst and Strecker, George E., Category Theory, Allen and Bacon, Inc. Boston (1973)