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 $y=c$ .^{[4]}
Example: The function $y(x)=2$ or just $y=2$ is the specific constant function where the output value is $c=2$. The domain of this function is the set of all real numbers ℝ. The codomain of this function is just {2}. 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)=2, y(−2.7)=2, y(π)=2,.... No matter what value of x is input, the output is "2".
Real-world example: A store where every item is sold for the price of 1 dollar.
The graph of the constant function $y=c$ is a horizontal line in the plane that passes through the point $(0,c)$.^{[5]}
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.^{[6]}
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.^{[7]} 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.^{[8]}
Example: Given the constant function $y(x)=-{\sqrt {2}}$ . The derivative of y is the identically zero function $y'(x)=(x\mapsto -{\sqrt {2}})'=0$ .
Every set X is isomorphic to the set of constant functions into it. For each element x and any set Y, there is a unique function ${\tilde {x}}:Y\rightarrow X$ such that ${\tilde {x}}(y)=x$ for all $y\in Y$. Conversely, if a function $f:Y\rightarrow X$ satisfies $f(y)=f(y')$ for all $y,y'\in Y$, $f$ is by definition a constant function.
As a corollary, the one-point set is a generator in the category of sets.
Every set $X$ is canonically isomorphic to the function set $X^{1}$, or hom set$\operatorname {hom} (1,X)$ in the category of sets, where 1 is the one-point set. Because of this, and the adjunction between cartesian products and hom in the category of sets (so there is a canonical isomorphism between functions of two variables and functions of one variable valued in functions of another (single) variable, $\operatorname {hom} (X\times Y,Z)\cong \operatorname {hom} (X(\operatorname {hom} (Y,Z))$) the category of sets is a closed monoidal category with the cartesian product of sets as tensor product and the one-point set as tensor unit. In the isomorphisms $\lambda :1\times X\cong X\cong X\times 1:\rho$natural in X, the left and right unitors are the projections $p_{1}$ and $p_{2}$ the ordered pairs$(*,x)$ and $(x,*)$ respectively to the element $x$, where $*$ is the unique point in the one-point set.