Constant function

In mathematics, a constant function is a function whose output value is the same for every input value. For example, the function formula_1 is a constant function because the value of  formula_2  is 4 regardless of the input value formula_3 (see image).
Basic properties.
Formally, a constant function "f"("x"):R→R has the form  formula_4. Usually we write formula_5  or just  formula_6. 
Example: The function  formula_7  or just  formula_8  is the specific constant function where the output value is  formula_9. 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".
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:  formula_12 .
Example:  formula_13  is a constant function. The derivative of "y" is the identically zero function   formula_14  .
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 "x"0"y" 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 on English Wikipedia