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 y(x) = 4 is a constant function because the value of y(x) is 4 regardless of the input value x (see image).

Basic properties

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, and so on. 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 where c is nonzero. 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: . 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 . 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.


  1. ^ Tanton, James (2005). Encyclopedia of Mathematics. Facts on File, New York. p. 94. ISBN 0-8160-5124-0.
  2. ^ C.Clapham, J.Nicholson (2009). "Oxford Concise Dictionary of Mathematics, Constant Function" (PDF). Addison-Wesley. p. 175. Retrieved January 12, 2014.
  3. ^ Weisstein, Eric (1999). CRC Concise Encyclopedia of Mathematics. CRC Press, London. p. 313. ISBN 0-8493-9640-9.
  4. ^ Weisstein, Eric W. "Constant Function". Retrieved 2020-07-27.
  5. ^ Dawkins, Paul (2007). "College Algebra". Lamar University. p. 224. Retrieved January 12, 2014.
  6. ^ 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.
  7. ^ Dawkins, Paul (2007). "Derivative Proofs". Lamar University. Retrieved January 12, 2014.
  8. ^ "Zero Derivative implies Constant Function". Retrieved January 12, 2014.
  9. ^ Leinster, Tom (27 Jun 2011). "An informal introduction to topos theory". arXiv:1012.5647 [math.CT].