Constant function

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 
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: formula_1. The converse is also true. Namely, if "y"′("x")=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.