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Constant function
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 [math]y(x) = 4[/math] is a constant function because the value of [math]y(x)[/math] is 4 regardless of the input value [math]x[/math] (see image).
Contents
Basic properties
Formally, a constant function f(x):R→R has the form [math]f(x)=c[/math]. Usually we write [math]y(x)=c[/math] or just [math]y=c[/math].
 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 [math]y(x)=4[/math] or just [math]y=4[/math] is the specific constant function where the output value is [math]c=4[/math]. 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 [math]y=c[/math] is a horizontal line in the plane that passes through the point [math](0,c)[/math].^{[4]}
 If c≠0, the constant function y=c is a polynomial in one variable x of degree zero.
 The yintercept of this function is the point (0,c).
 This function has no xintercept. That is, it has no root or zero. It never crosses the xaxis.
 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 xaxis in the plane.^{[5]}
 A constant function is an even function so the yaxis 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: [math](c)'=0[/math] .
Example: [math]y(x)=\sqrt{2}[/math] is a constant function. The derivative of y is the identically zero function [math]y'(x)=(\sqrt{2})'=0[/math] .
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:
 [math]y(x)=c \,\,\, \Leftrightarrow \,\,\, y'(x)=0 \,, \,\,\forall x \in \mathbb{R}[/math]
Generalization
A function f : A → B is a constant function if f(a) = f(b) for every a and b in A.^{[8]}
Examples
Realworld 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 : A → B where A={X,Y,Z,W} and B={1,2,3} and f(a)=3 for every a∈A. 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 3dimensional 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
Related pages
References
 ↑ Tanton, James (2005). Encyclopedia of Mathematics. Facts on File, New York. p. 94. . (in English)
 ↑ C.Clapham, J.Nicholson (2009). "Oxford Concise Dictionary of Mathematics, Constant Function" (in English). AddisonWesley. p. 175. http://web.cortland.edu/matresearch/OxfordDictionaryMathematics.pdf. Retrieved January 2014.
 ↑ Weisstein, Eric (1999). CRC Concise Encyclopedia of Mathematics. CRC Press, London. p. 313. . (in English)
 ↑ Dawkins, Paul (2007). "College Algebra" (in English). Lamar University. p. 224. http://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx. Retrieved January 2014.
 ↑ Carter, John A.; Cuevas, Gilbert J.; Holliday, Berchie; Marks, Daniel; McClure, Melissa S.publisher=Glencoe/McGrawHill School Pub Co (2005). Advanced Mathematical Concepts  Precalculus with Applications, Student Edition 1. p. 22. . (in English)
 ↑ Dawkins, Paul (2007). "Derivative Proofs" (in English). Lamar University. http://tutorial.math.lamar.edu/Classes/CalcI/DerivativeProofs.aspx. Retrieved January 2014.
 ↑ "Zero Derivative implies Constant Function" (in English). http://www.proofwiki.org/wiki/Zero_Derivative_implies_Constant_Function. Retrieved January 2014.
 ↑ "Constant Function" (in English). http://planetmath.org/ConstantFunction.html. Retrieved January 2014.
Other websites
 "Constant Function" (in english). From MathWorldA Wolfram Web Resource. http://mathworld.wolfram.com/ConstantFunction.html. Retrieved January 2014.
