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In mathematics, a constant function is a function whose (output) value is the same for every input value.^{[1]}^{[2]}^{[3]} For example, the function y(x) = 4 is a constant function because the value of y(x) is 4 regardless of the input value x (see image).
As a real-valued function of a real-valued argument, a constant function has the general form y(x)=c or just y=c .
The graph of the constant function y=c is a horizontal line in the plane that passes through the point (0,c).^{[4]}
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 \neq 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 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.^{[5]}
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.^{[6]} This is often written: (c)'=0 . The converse is also true. Namely, if y'(x)=0 for all real numbers x, then y(x) is a constant function.^{[7]}
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.
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