Constant function

In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function y ( x ) = 4 {displaystyle y(x)=4} is a constant function because the value of   y ( x ) {displaystyle y(x)}  is 4 regardless of the input value x {displaystyle x} (see image). In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function y ( x ) = 4 {displaystyle y(x)=4} is a constant function because the value of   y ( x ) {displaystyle y(x)}  is 4 regardless of the input value x {displaystyle x} (see image). As a real-valued function of a real-valued argument, a constant function has the general form   y ( x ) = c {displaystyle y(x)=c}  or just   y = c {displaystyle y=c}  . The graph of the constant function y = c {displaystyle y=c} is a horizontal line in the plane that passes through the point ( 0 , c ) {displaystyle (0,c)} . 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 f ( x ) = c , c ≠ 0 {displaystyle f(x)=c,,,,c eq 0}  . This function has no intersection point with the x-axis, that is, it has no root (zero). On the other hand, the polynomial   f ( x ) = 0 {displaystyle 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. 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:   ( c ) ′ = 0 {displaystyle (c)'=0}  . The converse is also true. Namely, if y'(x)=0 for all real numbers x, then y(x) is a constant function. 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.