In mathematics, a constant function is a function whose (output) value is the same for every input value.

Basic properties

An example of a constant function is y(x) = 4, because the value of y(x) is 4 regardless of the input value x.

As a real-valued function of a real-valued argument, a constant function has the general form y(x) = c or just y = c. For example, the function y(x) = 4 is the specific constant function where the output value is c = 4. The domain of this function is the set of all real numbers. The image of this function is the singleton set {4}. 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) = 4, y(−2.7) = 4, y(π) = 4, and so on. No matter what value of x is input, the output is 4.[1]

The graph of the constant function y = c is a horizontal line in the plane that passes through the point (0, c).[2] In the context of a polynomial in one variable x, the constant function is called non-zero constant function because it is a polynomial of degree 0, and its general form is f(x) = c, where c is nonzero. This function has no intersection point with the x-axis, meaning 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.[3] Its graph is symmetric with respect to the y-axis, and therefore a constant function is an even function.[4]

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.[5] This is often written: . The converse is also true. Namely, if y′(x) = 0 for all real numbers x, then y is a constant function.[6] For 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

  1. ^ Tanton, James (2005). Encyclopedia of Mathematics. Facts on File, New York. p. 94. ISBN 0-8160-5124-0.
  2. ^ Dawkins, Paul (2007). "College Algebra". Lamar University. p. 224. Retrieved January 12, 2014.
  3. ^ Carter, John A.; Cuevas, Gilbert J.; Holliday, Berchie; Marks, Daniel; McClure, Melissa S. (2005). "1". Advanced Mathematical Concepts - Pre-calculus with Applications, Student Edition (1 ed.). Glencoe/McGraw-Hill School Pub Co. p. 22. ISBN 978-0078682278.
  4. ^ Young, Cynthia Y. (2021). Precalculus (3rd ed.). p. 122.
  5. ^ Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007). Calculus (9th ed.). Pearson Prentice Hall. p. 107. ISBN 978-0131469686.
  6. ^ "Zero Derivative implies Constant Function". Retrieved January 12, 2014.
  7. ^ Leinster, Tom (27 Jun 2011). "An informal introduction to topos theory". arXiv:1012.5647 [math.CT].