Tarski's theorem

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Constant function y=4

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 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    .

Example: The function    or just    is the specific constant function where the output value is  . 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 euro.

The graph of the constant function is a horizontal line in the plane that passes through the point .[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  . 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.[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:   . The converse is also true. Namely, if y'(x)=0 for all real numbers x, then y(x) is a constant function.[7]

Example: Given the constant function    . The derivative of y is the identically zero 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.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

  • Herrlich, Horst and Strecker, George E., Category Theory, Allen and Bacon, Inc. Boston (1973)

External links

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  3. 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Name: Dario Ellis
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  4. Template:Cite web
  5. 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Name: Dario Ellis
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  6. Template:Cite web
  7. Template:Cite web