In mathematics, a constant function is a function whose (output) value is the same for every input value.

Basic properties

[[Image:wiki constant function 175 200.png|270px|right|thumb|An example of a constant function is y(x) = 4 , because the value of y(x) is 4 regardless of the input value x.]] As a real-valued function of a real-valued argument, a constant function has the general form y(x) = c or just For example, the function y(x) = 4 is the specific constant function where the output value is c = 4 . The domain of this function is the set of all real numbers. The image of this function is the singleton set . 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) = 4 , y(−2.7) = 4 , y(π) = 4 , and so on. No matter what value of x is input, the output is 4 . The graph of the constant function y = c is a horizontal line in the plane that passes through the point (0, c) . In the context of a polynomial in one variable x , the constant function is called non-zero constant function because it is a polynomial of degree 0, and its general form is f(x) = c , where c is nonzero. This function has no intersection point with the x -axis, meaning 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. Its graph is symmetric with respect to the y -axis, and therefore a constant function is an even function. 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 y′(x) = 0 for all real numbers x , then y is a constant function. For 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. X is a left zero of the full transformation monoid on X , which implies that it is also idempotent. X , every set Y is isomorphic to the set of constant functions in X \to Y. For any X and each element y in Y , there is a unique function such that for all x \in X. Conversely, if a function f: X \to Y satisfies for all x, x' \in X, f is by definition a constant function. X ]], the left and right unitors are the projections p_1 and p_2 the ordered pairs (*, x) and (x, *) respectively to the element x, where * is the unique point in the one-point set. A function on a connected set is locally constant if and only if it is constant.