Constant function


In mathematics, a constant function is a function whose 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 .

Basic properties

As a real-valued function of a real-valued argument, a constant function has the general form or just .
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 . This function has no intersection point with the x-axis, that is, it has no root. On the other hand, the polynomial is the identically zero function. It is the 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 y'=0 for all real numbers x, then y is a constant 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.
A function on a connected set is locally constant if and only if it is constant.