wikiwand.com

Constant function - Wikiwand

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

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 y = c. 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 {4}. 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.^[1]

The graph of the constant function y = c is a horizontal line in the plane that passes through the point (0, c).^[2] 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.^[3] Its graph is symmetric with respect to the y-axis, and therefore a constant function is an even function.^[4]

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.^[5] This is often written: $(x\mapsto c)'=0$ . The converse is also true. Namely, if y′(x) = 0 for all real numbers x, then y is a constant function.^[6] For example, given the constant function $y(x)=-{\sqrt {2}}$ . The derivative of y is the identically zero function $y'(x)=\left(x\mapsto -{\sqrt {2}}\right)'=0$ .

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.

A function on a connected set is locally constant if and only if it is constant.

Loading related searches...