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 As a real-valued function of a real-valued argument, a constant function has the general form or just Example: The function or just is the specific constant function where the output value is 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 and so on. 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 dollar. The graph of the constant function is a horizontal line in the plane that passes through the point 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 where is nonzero. This function has no intersection point with the x-axis, that is, it has no root (zero). On the other hand, the polynomial 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. 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. This is often written: . The converse is also true. Namely, if for all real numbers x, then y is a constant function. Example: Given the constant function The derivative of y is the identically zero function 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. Every constant function whose domain and codomain are the same set X is a left zero of the full transformation monoid on X, which implies