In mathematics, a constant function is a function whose output value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image).
Basic properties
Formally, a constant function f(x):R→R has the form . Usually we write or just . 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 or just is the specific constant function where the output value is . 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 is a horizontal line in the plane that passes through the point . 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. 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. This is often written: .
Example: is a constant function. The derivative of y is the identically zero function .
The converse (opposite) is also true. That is, if the derivative of a function is zero everywhere, then the function is a constant function.
Mathematically we write these two statements:
Generalization
A function f : A → B is a constant function if f(a) = f(b) for every a and b in A.
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 : 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 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.
Other properties
There are other properties of constant functions. See Constant function on English Wikipedia